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MODERN ANTENNA DESIGN Second Edition

THOMAS A. MILLIGAN

IEEE PRESS

A JOHN WILEY & SONS, INC., PUBLICATION

MODERN ANTENNA DESIGN

MODERN ANTENNA DESIGN Second Edition

THOMAS A. MILLIGAN

IEEE PRESS

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright  2005 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Milligan, Thomas A. Modern antenna design / by Thomas A. Milligan.—2nd ed. p. cm. Includes bibliographical references and index. ISBN-13 978-0-471-45776-3 (cloth) ISBN-10 0-471-45776-0 (cloth) 1. Antennas (Electronics)–Design and construction. I. Title. TK7871.6.M54 2005 621.382 4—dc22 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1

2004059098

To Mary, Jane, and Margaret

CONTENTS

Preface 1 Properties of Antennas

xv 1

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8

Antenna Radiation, 2 Gain, 3 Effective Area, 6 Path Loss, 6 Radar Range Equation and Cross Section, 7 Why Use an Antenna? 9 Directivity, 10 Directivity Estimates, 11 1-8.1 Pencil Beam, 11 1-8.2 Butterﬂy or Omnidirectional Pattern, 13 1-9 Beam Efﬁciency, 16 1-10 Input-Impedance Mismatch Loss, 17 1-11 Polarization, 18 1-11.1 Circular Polarization Components, 19 1-11.2 Huygens Source Polarization, 21 1-11.3 Relations Between Bases, 22 1-11.4 Antenna Polarization Response, 23 1-11.5 Phase Response of Rotating Antennas, 25 1-11.6 Partial Gain, 26 1-11.7 Measurement of Circular Polarization Using Amplitude Only, 26 1-12 Vector Effective Height, 27 1-13 Antenna Factor, 29 1-14 Mutual Coupling Between Antennas, 29 1.15 Antenna Noise Temperature, 30 vii

viii

CONTENTS

1-16 Communication Link Budget and Radar Range, 35 1-17 Multipath, 36 1-18 Propagation Over Soil, 37 1-19 Multipath Fading, 39 References, 40 2 Radiation Structures and Numerical Methods 2-1 Auxiliary Vector Potentials, 43 2-1.1 Radiation from Electric Currents, 44 2-1.2 Radiation from Magnetic Currents, 49 2-2 Apertures: Huygens Source Approximation, 51 2-2.1 Near- and Far-Field Regions, 55 2-2.2 Huygens Source, 57 2-3 Boundary Conditions, 57 2-4 Physical Optics, 59 2-4.1 Radiated Fields Given Currents, 59 2-4.2 Applying Physical Optics, 60 2-4.3 Equivalent Currents, 65 2-4.4 Reactance Theorem and Mutual Coupling, 66 2-5 Method of Moments, 67 2-5.1 Use of the Reactance Theorem for the Method of Moments, 68 2-5.2 General Moments Method Approach, 69 2-5.3 Thin-Wire Moment Method Codes, 71 2-5.4 Surface and Volume Moment Method Codes, 71 2-5.5 Examples of Moment Method Models, 72 2-6 Finite-Difference Time-Domain Method, 76 2-6.1 Implementation, 76 2-6.2 Central Difference Derivative, 77 2-6.3 Finite-Difference Maxwell’s Equations, 77 2-6.4 Time Step for Stability, 79 2-6.5 Numerical Dispersion and Stability, 80 2-6.6 Computer Storage and Execution Times, 80 2-6.7 Excitation, 81 2-6.8 Waveguide Horn Example, 83 2-7 Ray Optics and the Geometric Theory of Diffraction, 84 2-7.1 Fermat’s Principle, 85 2-7.2 H -Plane Pattern of a Dipole Located Over a Finite Strip, 85 2-7.3 E-Plane Pattern of a Rectangular Horn, 87 2-7.4 H -Plane Pattern of a Rectangular Horn, 89 2-7.5 Amplitude Variations Along a Ray, 90 2-7.6 Extra Phase Shift Through Caustics, 93 2-7.7 Snell’s Laws and Reﬂection, 93 2-7.8 Polarization Effects in Reﬂections, 94 2-7.9 Reﬂection from a Curved Surface, 94 2-7.10 Ray Tracing, 96

42

CONTENTS

2-7.11 2-7.12 2-7.13 2-7.14 2-7.15 References, 100

ix

Edge Diffraction, 96 Slope Diffraction, 98 Corner Diffraction, 99 Equivalent Currents, 99 Diffraction from Curved Surfaces, 99

3 Arrays

102

3-1 Two-Element Array, 104 3-2 Linear Array of N Elements, 109 3-3 Hansen and Woodyard End-Fire Array, 114 3-4 Phased Arrays, 115 3-5 Grating Lobes, 117 3-6 Multiple Beams, 118 3-7 Planar Array, 120 3-8 Grating Lobes in Planar Arrays, 125 3-9 Mutual Impedance, 127 3-10 Scan Blindness and Array Element Pattern, 127 3-11 Compensating Array Feeding for Mutual Coupling, 128 3-12 Array Gain, 129 3-13 Arrays Using Arbitrarily Oriented Elements, 133 References, 135 4 Aperture Distributions and Array Synthesis 4-1 Amplitude Taper and Phase Error Efﬁciencies, 137 4-1.1 Separable Rectangular Aperture Distributions, 139 4-1.2 Circularly Symmetrical Distributions, 140 4-2 Simple Linear Distributions, 140 4-3 Taylor One-Parameter Linear Distribution, 144 4-4 Taylor n Line Distribution, 147 4-5 Taylor Line Distribution with Edge Nulls, 152 4-6 Elliott’s Method for Modiﬁed Taylor Distribution and Arbitrary Sidelobes, 155 4-7 Bayliss Line-Source Distribution, 158 4-8 Woodward Line-Source Synthesis, 162 4-9 Schelkunoff’s Unit-Circle Method, 164 4-10 Dolph–Chebyshev Linear Array, 170 4-11 Villeneuve Array Synthesis, 172 4-12 Zero Sampling of Continuous Distributions, 173 4-13 Fourier Series Shaped-Beam Array Synthesis, 175 4-14 Orchard Method of Array Synthesis, 178 4-15 Series-Fed Array and Traveling-Wave Feed Synthesis, 188 4-16 Circular Apertures, 191 4-17 Circular Gaussian Distribution, 194 4-18 Hansen Single-Parameter Circular Distribution, 195 4-19 Taylor Circular-Aperture Distribution, 196 4-20 Bayliss Circular-Aperture Distribution, 200

136

x

CONTENTS

4-21 4-22 4-23 4-24 4-25

Planar Arrays, 202 Convolution Technique for Planar Arrays, 203 Aperture Blockage, 208 Quadratic Phase Error, 211 Beam Efﬁciency of Circular Apertures with Axisymmetric Distribution, 214 References, 215 5 Dipoles, Slots, and Loops 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5-10 5-11 5-12 5-13 5-14 5-15

5-16 5-17 5-18 5-19 5-20 5-21 5-22 5-23 5-24 5-25 5-26

Standing-Wave Currents, 218 Radiation Resistance (Conductance), 220 Babinet–Booker Principle, 222 Dipoles Located Over a Ground Plane, 223 Dipole Mounted Over Finite Ground Planes, 225 Crossed Dipoles for Circular Polarization, 231 Super Turnstile or Batwing Antenna, 234 Corner Reﬂector, 237 Monopole, 242 Sleeve Antenna, 242 Cavity-Mounted Dipole Antenna, 245 Folded Dipole, 247 Shunt Feeding, 248 Discone Antenna, 249 Baluns, 251 5-15.1 Folded Balun, 252 5-15.2 Sleeve or Bazooka Baluns, 253 5-15.3 Split Coax Balun, 255 5-15.4 Half-Wavelength Balun, 256 5-15.5 Candelabra Balun, 256 5-15.6 Ferrite Core Baluns, 256 5-15.7 Ferrite Candelabra Balun, 258 5-15.8 Transformer Balun, 258 5-15.9 Split Tapered Coax Balun, 259 5-15.10 Natural Balun, 260 Small Loop, 260 Alford Loop, 261 Resonant Loop, 263 Quadriﬁlar Helix, 264 Cavity-Backed Slots, 266 Stripline Series Slots, 266 Shallow-Cavity Crossed-Slot Antenna, 269 Waveguide-Fed Slots, 270 Rectangular-Waveguide Wall Slots, 271 Circular-Waveguide Slots, 276 Waveguide Slot Arrays, 278 5-26.1 Nonresonant Array, 279 5-26.2 Resonant Array, 282

217

CONTENTS

xi

5-26.3 Improved Design Methods, 282 References, 283 6 Microstrip Antennas

285

6-1 Microstrip Antenna Patterns, 287 6-2 Microstrip Patch Bandwidth and Surface-Wave Efﬁciency, 293 6-3 Rectangular Microstrip Patch Antenna, 299 6-4 Quarter-Wave Patch Antenna, 310 6-5 Circular Microstrip Patch, 313 6-6 Circularly Polarized Patch Antennas, 316 6-7 Compact Patches, 319 6-8 Directly Fed Stacked Patches, 323 6-9 Aperture-Coupled Stacked Patches, 325 6-10 Patch Antenna Feed Networks, 327 6-11 Series-Fed Array, 329 6-12 Microstrip Dipole, 330 6-13 Microstrip Franklin Array, 332 6-14 Microstrip Antenna Mechanical Properties, 333 References, 334 7 Horn Antennas

336

7-1 Rectangular Horn (Pyramidal), 337 7-1.1 Beamwidth, 341 7-1.2 Optimum Rectangular Horn, 343 7-1.3 Designing to Given Beamwidths, 346 7-1.4 Phase Center, 347 7-2 Circular-Aperture Horn, 348 7-2.1 Beamwidth, 350 7-2.2 Phase Center, 352 7-3 Circular (Conical) Corrugated Horn, 353 7-3.1 Scalar Horn, 357 7-3.2 Corrugation Design, 357 7-3.3 Choke Horns, 358 7-3.4 Rectangular Corrugated Horns, 359 7-4 Corrugated Ground Plane, 359 7-5 Gaussian Beam, 362 7-6 Ridged Waveguide Horns, 365 7-7 Box Horn, 372 7-8 T-Bar-Fed Slot Antenna, 374 7-9 Multimode Circular Horn, 376 7-10 Biconical Horn, 376 References, 378 8 Reﬂector Antennas 8-1 Paraboloidal Reﬂector Geometry, 381 8-2 Paraboloidal Reﬂector Aperture Distribution Losses, 383

380

xii

CONTENTS

8-3 8-4 8-5 8-6 8-7 8-8 8-9 8-10 8-11 8-12 8-13

Approximate Spillover and Amplitude Taper Trade-offs, 385 Phase Error Losses and Axial Defocusing, 387 Astigmatism, 389 Feed Scanning, 390 Random Phase Errors, 393 Focal Plane Fields, 396 Feed Mismatch Due to the Reﬂector, 397 Front-to-Back Ratio, 399 Offset-Fed Reﬂector, 399 Reﬂections from Conic Sections, 405 Dual-Reﬂector Antennas, 408 8-13.1 Feed Blockage, 410 8-13.2 Diffraction Loss, 413 8-13.3 Cassegrain Tolerances, 414 8-14 Feed and Subreﬂector Support Strut Radiation, 416 8-15 Gain/Noise Temperature of a Dual Reﬂector, 421 8-16 Displaced-Axis Dual Reﬂector, 421 8-17 Offset-Fed Dual Reﬂector, 424 8-18 Horn Reﬂector and Dragonian Dual Reﬂector, 427 8-19 Spherical Reﬂector, 429 8-20 Shaped Reﬂectors, 432 8-20.1 Cylindrical Reﬂector Synthesis, 433 8-20.2 Circularly Symmetrical Reﬂector Synthesis, 434 8-20.3 Doubly Curved Reﬂector for Shaped Beams, 437 8-20.4 Dual Shaped Reﬂectors, 439 8-21 Optimization Synthesis of Shaped and Multiple-Beam Reﬂectors, 442 References, 443 9 Lens Antennas

447

9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8

Single Refracting Surface Lenses, 448 Zoned Lenses, 451 General Two-Surface Lenses, 454 Single-Surface or Contact Lenses, 459 Metal Plate Lenses, 461 Surface Mismatch and Dielectric Losses, 463 Feed Scanning of a Hyperboloidal Lens, 464 Dual-Surface Lenses, 465 9-8.1 Coma-Free Axisymmetric Dielectric Lens, 466 9-8.2 Speciﬁed Aperture Distribution Axisymmetric Dielectric Lens, 468 9-9 Bootlace Lens, 470 9-10 Luneburg Lens, 472 References, 472 10

Traveling-Wave Antennas 10-1 General Traveling Waves, 475

474

CONTENTS

xiii

10-1.1 Slow Wave, 478 10-1.2 Fast Waves (Leaky Wave Structure), 480 10-2 Long Wire Antennas, 481 10-2.1 Beverage Antenna, 481 10-2.2 V Antenna, 482 10-2.3 Rhombic Antenna, 483 10-3 Yagi–Uda Antennas, 485 10-3.1 Multiple-Feed Yagi–Uda Antennas, 492 10-3.2 Resonant Loop Yagi–Uda Antennas, 495 10-4 Corrugated Rod (Cigar) Antenna, 497 10-5 Dielectric Rod (Polyrod) Antenna, 499 10-6 Helical Wire Antenna, 502 10-6.1 Helical Modes, 503 10-6.2 Axial Mode, 504 10-6.3 Feed of a Helical Antenna, 506 10-6.4 Long Helical Antenna, 507 10-6.5 Short Helical Antenna, 508 10-7 Short Backﬁre Antenna, 509 10-8 Tapered Slot Antennas, 512 10-9 Leaky Wave Structures, 516 References, 518 11 Frequency-Independent Antennas Spiral 11-1 11-2 11-3 11-4 11-5

Antennas, 522 Modal Expansion of Antenna Patterns, 524 Archimedean Spiral, 526 Equiangular Spiral, 527 Pattern Analysis of Spiral Antennas, 530 Spiral Construction and Feeding, 535 11-5.1 Spiral Construction, 535 11-5.2 Balun Feed, 536 11-5.3 Inﬁnite Balun, 538 11-5.4 Beamformer and Coaxial Line Feed, 538 11-6 Spiral and Beamformer Measurements, 538 11-7 Feed Network and Antenna Interaction, 540 11-8 Modulated Arm Width Spiral, 541 11-9 Conical Log Spiral Antenna, 543 11-10 Mode 2 Conical Log Spiral Antenna, 549 11-11 Feeding Conical Log Spirals, 550 Log-Periodic Antennas, 550 11-12 Log-Periodic Dipole Antenna, 551 11-12.1 Feeding a Log-Periodic Dipole Antenna, 556 11-12.2 Phase Center, 558 11-12.3 Elevation Angle, 559 11-12.4 Arrays of Log-Periodic Dipole Antennas, 560 11-13 Other Log-Periodic Types, 561 11-14 Log-Periodic Antenna Feeding Paraboloidal Reﬂector, 563

521

xiv

CONTENTS

11-15 V Log-Periodic Array, 567 11-16 Cavity-Backed Planar Log-Periodic Antennas, 569 References, 571 12

Phased Arrays

573

12-1 12-2 12-3 12-4

Fixed Phase Shifters (Phasers), 574 Quantization Lobes, 578 Array Errors, 580 Nonuniform and Random Element Existence Arrays, 582 12-4.1 Linear Space Tapered Array, 582 12-4.2 Circular Space Tapered Array, 584 12-4.3 Statistically Thinned Array, 587 12-5 Array Element Pattern, 588 12-6 Feed Networks, 590 12-6.1 Corporate Feed, 590 12-6.2 Series Feed, 592 12-6.3 Variable Power Divider and Phase Shifter, 592 12-6.4 Butler Matrix, 594 12-6.5 Space Feeding, 596 12-6.6 Tapered Feed Network with Uniform-Amplitude Subarrays, 597 12-7 Pattern Null Formation in Arbitrary Array, 599 12-8 Phased Array Application to Communication Systems, 601 12-9 Near-Field Measurements on Phased Arrays, 602 References, 604 Index

607

PREFACE

I wrote this book from my perspective as a designer in industry, primarily for other designers and users of antennas. On occasion I have prepared and taught antenna courses, for which I developed a systematic approach to the subject. For the last decade I have edited the “Antenna Designer’s Notebook” column in the IEEE antenna magazine. This expanded edition adds a combination of my own design notebook and the many other ideas provided to me by others, leading to this collection of ideas that I think designers should know. The book contains a systematic approach to the subject. Every author would like to be read from front to back, but my own career assignments would have caused to me to jump around in this book. Nevertheless, Chapter 1 covers those topics that every user and designer should know. Because I deal with complete antenna design, which includes mounting the antenna, included are the effects of nearby structures and how they can be used to enhance the response. We all study ideal antennas ﬂoating in free space to help us understand the basics, but the real world is a little different. Instead of drawing single line graphs of common relationships between two parameters, I generated scales for calculations that I perform over and over. I did not supply a set of computer programs because I seldom use collections supplied by others, and younger engineers ﬁnd my programs quaint, as each generation learns a different computer language. You’ll learn by writing your own. IEEE Antennas and Propagation Society’s digital archive of all material published from 1952 to 2000 has changed our approach to research. I have not included extensive bibliographies, because I believe that it is no longer necessary. The search engine of the archive can supply an exhaustive list. I referred only to papers that I found particularly helpful. Complete sets of the transactions are available in libraries, whereas the wealth of information in the archive from conferences was not. I have started mining this information, which contains many useful design ideas, and have incorporated some of them in this book. In this ﬁeld, 40-year-old publications are still useful and we should not reinvent methods. Many clever ideas from industry are usually published xv

xvi

PREFACE

only once, if at all, and personally, I’ll be returning to this material again and again, because all books have limited space. As with the ﬁrst edition, I enjoyed writing this book because I wanted to express my point of view of a rewarding ﬁeld. Although the amount of information available is overwhelming and the mathematics describing it can cloud the ideas, I hope my explanations help you develop new products or use old ones. I would like to thank all the authors who taught me this subject by sharing their ideas, especially those working in industry. On a personal note I thank the designers at Lockheed Martin, who encouraged me and reviewed material while I wrote: in particular, Jeannette McDonnell, Thomas Cencich, Donald Huebner, and Julie Huffman. THOMAS A. MILLIGAN

1 PROPERTIES OF ANTENNAS

One approach to an antenna book starts with a discussion of how antennas radiate. Beginning with Maxwell’s equations, we derive electromagnetic waves. After that lengthy discussion, which contains a lot of mathematics, we discuss how these waves excite currents on conductors. The second half of the story is that currents radiate and produce electromagnetic waves. You may already have studied that subject, or if you wish to further your background, consult books on electromagnetics. The study of electromagnetics gives insight into the mathematics describing antenna radiation and provides the rigor to prevent mistakes. We skip the discussion of those equations and move directly to practical aspects. It is important to realize that antennas radiate from currents. Design consists of controlling currents to produce the desired radiation distribution, called its pattern. In many situations the problem is how to prevent radiation from currents, such as in circuits. Whenever a current becomes separated in distance from its return current, it radiates. Simply stated, we design to keep the two currents close together, to reduce radiation. Some discussions will ignore the current distribution and instead, consider derived quantities, such as ﬁelds in an aperture or magnetic currents in a slot or around the edges of a microstrip patch. You will discover that we use any concept that provides insight or simpliﬁes the mathematics. An antenna converts bound circuit ﬁelds into propagating electromagnetic waves and, by reciprocity, collects power from passing electromagnetic waves. Maxwell’s equations predict that any time-varying electric or magnetic ﬁeld produces the opposite ﬁeld and forms an electromagnetic wave. The wave has its two ﬁelds oriented orthogonally, and it propagates in the direction normal to the plane deﬁned by the perpendicular electric and magnetic ﬁelds. The electric ﬁeld, the magnetic ﬁeld, and the direction of propagation form a right-handed coordinate system. The propagating wave ﬁeld intensity decreases by 1/R away from the source, whereas a static ﬁeld Modern Antenna Design, Second Edition, By Thomas A. Milligan Copyright  2005 John Wiley & Sons, Inc.

1

2

PROPERTIES OF ANTENNAS

drops off by 1/R 2 . Any circuit with time-varying ﬁelds has the capability of radiating to some extent. We consider only time-harmonic ﬁelds and use phasor notation with time dependence ej ωt . An outward-propagating wave is given by e−j (kR−ωt) , where k, the wave number, is given by 2π/λ. λ is the wavelength of the wave given by c/f , where c is the velocity of light (3 × 108 m/s in free space) and f is the frequency. Increasing the distance from the source decreases the phase of the wave. Consider a two-wire transmission line with ﬁelds bound to it. The currents on a single wire will radiate, but as long as the ground return path is near, its radiation will nearly cancel the other line’s radiation because the two are 180◦ out of phase and the waves travel about the same distance. As the lines become farther and farther apart, in terms of wavelengths, the ﬁelds produced by the two currents will no longer cancel in all directions. In some directions the phase delay is different for radiation from the current on each line, and power escapes from the line. We keep circuits from radiating by providing close ground returns. Hence, high-speed logic requires ground planes to reduce radiation and its unwanted crosstalk.

1-1 ANTENNA RADIATION Antennas radiate spherical waves that propagate in the radial direction for a coordinate system centered on the antenna. At large distances, spherical waves can be approximated by plane waves. Plane waves are useful because they simplify the problem. They are not physical, however, because they require inﬁnite power. The Poynting vector describes both the direction of propagation and the power density of the electromagnetic wave. It is found from the vector cross product of the electric and magnetic ﬁelds and is denoted S: S = E × H∗

W/m2

Root mean square (RMS) values are used to express the magnitude of the ﬁelds. H∗ is the complex conjugate of the magnetic ﬁeld phasor. The magnetic ﬁeld is proportional to the electric ﬁeld in the far ﬁeld. The constant of proportion is η, the impedance of free space (η = 376.73 ): |S| = S =

|E|2 η

W/m2

(1-1)

Because the Poynting vector is the vector product of the two ﬁelds, it is orthogonal to both ﬁelds and the triplet deﬁnes a right-handed coordinate system: (E, H, S). Consider a pair of concentric spheres centered on the antenna. The ﬁelds around the antenna decrease as 1/R, 1/R 2 , 1/R 3 , and so on. Constant-order terms would require that the power radiated grow with distance and power would not be conserved. For ﬁeld terms proportional to 1/R 2 , 1/R 3 , and higher, the power density decreases with distance faster than the area increases. The energy on the inner sphere is larger than that on the outer sphere. The energies are not radiated but are instead concentrated around the antenna; they are near-ﬁeld terms. Only the 1/R 2 term of the Poynting vector (1/R ﬁeld terms) represents radiated power because the sphere area grows as R 2 and

GAIN

3

gives a constant product. All the radiated power ﬂowing through the inner sphere will propagate to the outer sphere. The sign of the input reactance depends on the near-ﬁeld predominance of ﬁeld type: electric (capacitive) or magnetic (inductive). At resonance (zero reactance) the stored energies due to the near ﬁelds are equal. Increasing the stored ﬁelds increases the circuit Q and narrows the impedance bandwidth. Far from the antenna we consider only the radiated ﬁelds and power density. The power ﬂow is the same through concentric spheres: 4πR12 S1,avg = 4πR22 S2,avg The average power density is proportional to 1/R 2 . Consider differential areas on the two spheres at the same coordinate angles. The antenna radiates only in the radial direction; therefore, no power may travel in the θ or φ direction. Power travels in ﬂux tubes between areas, and it follows that not only the average Poynting vector but also every part of the power density is proportional to 1/R 2 : S1 R12 sin θ dθ dφ = S2 R22 sin θ dθ dφ Since in a radiated wave S is proportional to 1/R 2 , E is proportional to 1/R. It is convenient to deﬁne radiation intensity to remove the 1/R 2 dependence: U (θ, φ) = S(R, θ, φ)R 2

W/solid angle

Radiation intensity depends only on the direction of radiation and remains the same at all distances. A probe antenna measures the relative radiation intensity (pattern) by moving in a circle (constant R) around the antenna. Often, of course, the antenna rotates and the probe is stationary. Some patterns have established names. Patterns along constant angles of the spherical coordinates are called either conical (constant θ ) or great circle (constant φ). The great circle cuts when φ = 0◦ or φ = 90◦ are the principal plane patterns. Other named cuts are also used, but their names depend on the particular measurement positioner, and it is necessary to annotate these patterns carefully to avoid confusion between people measuring patterns on different positioners. Patterns are measured by using three scales: (1) linear (power), (2) square root (ﬁeld intensity), and (3) decibels (dB). The dB scale is used the most because it reveals more of the low-level responses (sidelobes). Figure 1-1 demonstrates many characteristics of patterns. The half-power beamwidth is sometimes called just the beamwidth. The tenth-power and null beamwidths are used in some applications. This pattern comes from a parabolic reﬂector whose feed is moved off the axis. The vestigial lobe occurs when the ﬁrst sidelobe becomes joined to the main beam and forms a shoulder. For a feed located on the axis of the parabola, the ﬁrst sidelobes are equal.

1-2 GAIN Gain is a measure of the ability of the antenna to direct the input power into radiation in a particular direction and is measured at the peak radiation intensity. Consider the

4

PROPERTIES OF ANTENNAS

Angle 3 dB

Half-power beamwidth

Pattern level

10 dB

Tenth-power beamwidth

20 dB

Sidelobes

Vestigial lobe 30 dB

Null beamwidth

40 dB

FIGURE 1-1

Antenna pattern characteristics.

power density radiated by an isotropic antenna with input power P0 at a distance R: S = P0 /4πR 2 . An isotropic antenna radiates equally in all directions, and its radiated power density S is found by dividing the radiated power by the area of the sphere 4πR 2 . The isotropic radiator is considered to be 100% efﬁcient. The gain of an actual antenna increases the power density in the direction of the peak radiation: P0 G |E|2 1 P0 Gη S= = (1-2) or |E| = = Sη 4πR 2 η R 4π Gain is achieved by directing the radiation away from other parts of the radiation sphere. In general, gain is deﬁned as the gain-biased pattern of the antenna: P0 G(θ, φ) 4πR 2 P0 G(θ, φ) U (θ, φ) = 4π S(θ, φ) =

power density radiation intensity

(1-3)

The surface integral of the radiation intensity over the radiation sphere divided by the input power P0 is a measure of the relative power radiated by the antenna, or the antenna efﬁciency: 2π π Pr G(θ, φ) sin θ dθ dφ = ηe = efﬁciency P0 4π 0 0

GAIN

5

where Pr is the radiated power. Material losses in the antenna or reﬂected power due to poor impedance match reduce the radiated power. In this book, integrals in the equation above and those that follow express concepts more than operations we perform during design. Only for theoretical simpliﬁcations of the real world can we ﬁnd closed-form solutions that would call for actual integration. We solve most integrals by using numerical methods that involve breaking the integrand into small segments and performing a weighted sum. However, it is helpful that integrals using measured values reduce the random errors by averaging, which improves the result. In a system the transmitter output impedance or the receiver input impedance may not match the antenna input impedance. Peak gain occurs for a receiver impedance conjugate matched to the antenna, which means that the resistive parts are the same and the reactive parts are the same magnitude but have opposite signs. Precision gain measurements require a tuner between the antenna and receiver to conjugate-match the two. Alternatively, the mismatch loss must be removed by calculation after the measurement. Either the effect of mismatches is considered separately for a given system, or the antennas are measured into the system impedance and mismatch loss is considered to be part of the efﬁciency. Example Compute the peak power density at 10 km of an antenna with an input power of 3 W and a gain of 15 dB. First convert dB gain to a ratio: G = 1015/10 = 31.62. The power spreads over the sphere area with radius 10 km or an area of 4π(104 )2 m2 . The power density is S=

(3 W)(31.62) = 75.5 nW/m2 4π × 108 m2

We calculate the electric ﬁeld intensity using Eq. (1-2): |E| = Sη = (75.5 × 10−9 )(376.7) = 5333 µV/m Although gain is usually relative to an isotropic antenna, some antenna gains are referred to a λ/2 dipole with an isotropic gain of 2.14 dB. If we approximate the antenna as a point source, we compute the electric ﬁeld radiated by using Eq. (1-2): e−j kR P0 G(θ, φ)η E(θ, φ) = (1-4) R 4π This requires only that the antenna be small compared to the radial distance R. Equation (1-4) ignores the direction of the electric ﬁeld, which we deﬁne as polarization. The units of the electric ﬁeld are volts/meter. We determine the far-ﬁeld pattern by multiplying Eq. (1-4) by R and removing the phase term e−j kR since phase has meaning only when referred to another point in the far ﬁeld. The far-ﬁeld electric ﬁeld Eff unit is volts: 2 P0 G(θ, φ)η 4π 1 Eff (θ, φ) (1-5) Eff (θ, φ) = or G(θ, φ) = 4π P0 η During analysis, we often normalize input power to √ 1 W and can compute gain easily from the electric ﬁeld by multiplying by a constant 4π/η = 0.1826374.

6

PROPERTIES OF ANTENNAS

1-3 EFFECTIVE AREA Antennas capture power from passing waves and deliver some of it to the terminals. Given the power density of the incident wave and the effective area of the antenna, the power delivered to the terminals is the product Pd = SAeff

(1-6)

For an aperture antenna such as a horn, parabolic reﬂector, or ﬂat-plate array, effective area is physical area multiplied by aperture efﬁciency. In general, losses due to material, distribution, and mismatch reduce the ratio of the effective area to the physical area. Typical estimated aperture efﬁciency for a parabolic reﬂector is 55%. Even antennas with inﬁnitesimal physical areas, such as dipoles, have effective areas because they remove power from passing waves. 1-4 PATH LOSS [1, p. 183] We combine the gain of the transmitting antenna with the effective area of the receiving antenna to determine delivered power and path loss. The power density at the receiving antenna is given by Eq. (1-3), and the received power is given by Eq. (1-6). By combining the two, we obtain the path loss: Pd A2 G1 (θ, φ) = Pt 4πR 2 Antenna 1 transmits, and antenna 2 receives. If the materials in the antennas are linear and isotropic, the transmitting and receiving patterns are identical (reciprocal) [2, p. 116]. When we consider antenna 2 as the transmitting antenna and antenna 1 as the receiving antenna, the path loss is Pd A1 G2 (θ, φ) = Pt 4πR 2 Since the responses are reciprocal, the path losses are equal and we can gather and eliminate terms: G1 G2 = = constant A1 A2 Because the antennas were arbitrary, this quotient must equal a constant. This constant was found by considering the radiation between two large apertures [3]: 4π G = 2 A λ

(1-7)

We substitute this equation into path loss to express it in terms of the gains or effective areas: Pd A1 A2 λ 2 = G1 G2 = 2 2 (1-8) Pt 4πR λR We make quick evaluations of path loss for various units of distance R and for frequency f in megahertz using the formula path loss(dB) = KU + 20 log(f R) − G1 (dB) − G2 (dB)

(1-9)

RADAR RANGE EQUATION AND CROSS SECTION

7

where KU depends on the length units: KU

Unit

km 32.45 nm 37.80 miles 36.58 m −27.55 ft −37.87 Example Compute the gain of a 3-m-diameter parabolic reﬂector at 4 GHz assuming 55% aperture efﬁciency. Gain is related to effective area by Eq. (1-7): G=

4πA λ2

We calculate the area of a circular aperture by A = π(D/2)2 . By combining these equations, we have πD 2 πDf 2 G= ηa = ηa (1-10) λ c where D is the diameter and ηa is the aperture efﬁciency. On substituting the values above, we obtain the gain:

3π(4 × 109 ) G= 0.3 × 109

2 (0.55) = 8685

(39.4 dB)

Example Calculate the path loss of a 50-km communication link at 2.2 GHz using a transmitter antenna with a gain of 25 dB and a receiver antenna with a gain of 20 dB. Path loss = 32.45 + 20 log[2200(50)] − 25 − 20 = 88.3 dB What happens to transmission between two apertures as the frequency is increased? If we assume that the effective area remains constant, as in a parabolic reﬂector, the transmission increases as the square of frequency: Pd A1 A2 1 A1 A2 f 2 = = = Bf 2 Pt R 2 λ2 R2 c where B is a constant for a ﬁxed range. The receiving aperture captures the same power regardless of frequency, but the gain of the transmitting antenna increases as the square of frequency. Hence, the received power also increases as frequency squared. Only for antennas, whose gain is a ﬁxed value when frequency changes, does the path loss increase as the square of frequency. 1-5 RADAR RANGE EQUATION AND CROSS SECTION Radar operates using a double path loss. The radar transmitting antenna radiates a ﬁeld that illuminates a target. These incident ﬁelds excite surface currents that also radiate

8

PROPERTIES OF ANTENNAS

to produce a second ﬁeld. These ﬁelds propagate to the receiving antenna, where they are collected. Most radars use the same antenna both to transmit the ﬁeld and to collect the signal returned, called a monostatic system, whereas we use separate antennas for bistatic radar. The receiving system cannot be detected in a bistatic system because it does not transmit and has greater survivability in a military application. We determine the power density illuminating the target at a range RT by using Eq. (1-2): PT GT (θ, φ) Sinc = (1-11) 4πRT2 The target’s radar cross section (RCS), the scattering area of the object, is expressed in square meters or dBm2 : 10 log(square meters). The RCS depends on both the incident and reﬂected wave directions. We multiply the power collected by the target with its receiving pattern by the gain of the effective antenna due to the currents induced: RCS = σ =

powerreﬂected Ps (θr , φr , θi , φi ) = power density incident PT GT /4πRT2

(1-12)

In a communication system we call Ps the equivalent isotropic radiated power (EIRP), which equals the product of the input power and the antenna gain. The target becomes the transmitting source and we apply Eq. (1-2) to ﬁnd the power density at the receiving antenna at a range RR from the target. Finally, the receiving antenna collects the power density with an effective area AR . We combine these ideas to obtain the power delivered to the receiver: AR PT GT σ (θr , φr , θi , φi ) Prec = SR AR = (4πRT2 )(4πRR2 ) We apply Eq. (1-7) to eliminate the effective area of the receiving antenna and gather terms to determine the bistatic radar range equation: Prec GT GR λ2 σ (θr , φr , θi , φi ) = PT (4π)3 RT2 RR2

(1-13)

We reduce Eq. (1-13) and collect terms for monostatic radar, where the same antenna is used for both transmitting and receiving: Prec G2 λ2 σ = PT (4π)3 R 4 Radar received power is proportional to 1/R 4 and to G2 . We ﬁnd the approximate RCS of a ﬂat plate by considering the plate as an antenna with an effective area. Equation (1-11) gives the power density incident on the plate that collects this power over an area AR : PC =

PT GT (θ, φ) AR 4πRT2

The power scattered by the plate is the power collected, PC , times the gain of the plate as an antenna, GP : Ps = PC GP =

PT GT (θi , φi ) AR GP (θr , φr ) 4πRT2

WHY USE AN ANTENNA?

9

This scattered power is the effective radiated power in a particular direction, which in an antenna is the product of the input power and the gain in a particular direction. We calculate the plate gain by using the effective area and ﬁnd the scattered power in terms of area: PT GT 4πA2R Ps = 4πRT2 λ2 We determine the RCS σ by Eq. (1-12), the scattered power divided by the incident power density: σ =

Ps 4πA2R GR (θi , φi )GR (θr , φr )λ2 = = λ2 4π PT GT /4πRT2

(1-14)

The right expression of Eq. (1-14) divides the gain into two pieces for bistatic scattering, where the scattered direction is different from the incident direction. Monostatic scattering uses the same incident and reﬂected directions. We can substitute any object for the ﬂat plate and use the idea of an effective area and its associated antenna gain. An antenna is an object with a unique RCS characteristic because part of the power received will be delivered to the antenna terminals. If we provide a good impedance match to this signal, it will not reradiate and the RCS is reduced. When we illuminate an antenna from an arbitrary direction, some of the incident power density will be scattered by the structure and not delivered to the antenna terminals. This leads to the division of antenna RCS into the antenna mode of reradiated signals caused by terminal mismatch and the structural mode, the ﬁelds reﬂected off the structure for incident power density not delivered to the terminals. 1-6 WHY USE AN ANTENNA? We use antennas to transfer signals when no other way is possible, such as communication with a missile or over rugged mountain terrain. Cables are expensive and take a long time to install. Are there times when we would use antennas over level ground? The large path losses of antenna systems lead us to believe that cable runs are better. Example Suppose that we must choose between using a low-loss waveguide run and a pair of antennas at 3 GHz. Each antenna has 10 dB of gain. The low-loss waveguide has only 19.7 dB/km loss. Table 1-1 compares losses over various distances. The waveguide link starts out with lower loss, but the antenna system soon overtakes it. When the path length doubles, the cable link loss also doubles in decibels, but an antenna link TABLE 1-1 Losses Over Distance Distance (km)

Waveguide Loss (dB)

Antenna Path Loss (dB)

2 4 6 10

39.4 78.8 118.2 197

88 94 97.6 102

10

PROPERTIES OF ANTENNAS

increases by only 6 dB. As the distance is increased, radiating between two antennas eventually has lower losses than in any cable. Example A 200-m outside antenna range was set up to operate at 2 GHz using a 2-mdiameter reﬂector as a source. The receiver requires a sample of the transmitter signal to phase-lock the local oscillator and signal at a 45-MHz difference. It was proposed to run an RG/U 115 cable through the power and control cable conduit, since the run was short. The cable loss was 36 dB per 100 m, giving a total cable loss of 72 dB. A 10-dB coupler was used on the transmitter to pick off the reference signal, so the total loss was 82 dB. Since the source transmitted 100 mW (20 dBm), the signal was −62 dBm at the receiver, sufﬁcient for phase lock. A second proposed method was to place a standard-gain horn (15 dB of gain) within the beam of the source on a small stand out of the way of the measurement and next to the receiver. If we assume that the source antenna had only 30% aperture efﬁciency, we compute gain from Eq. (1-10) (λ = 0.15 m): G=

2π 0.15

2 (0.3) = 526

(27.2 dB)

The path loss is found from Eq. (1-9) for a range of 0.2 km: 32.45 + 20 log[2000(0.2)] − 27.2 − 15 = 42.3 dB The power delivered out of the horn is 20 dBm − 42.3 dB = −22.3 dBm. A 20-dB attenuator must be put on the horn to prevent saturation of the receiver (−30 dBm). Even with a short run, it is sometimes better to transmit the signal between two antennas instead of using cables. 1-7 DIRECTIVITY Directivity is a measure of the concentration of radiation in the direction of the maximum: maximum radiation intensity Umax directivity = = (1-15) average radiation intensity U0 Directivity and gain differ only by the efﬁciency, but directivity is easily estimated from patterns. Gain—directivity times efﬁciency—must be measured. The average radiation intensity can be found from a surface integral over the radiation sphere of the radiation intensity divided by 4π, the area of the sphere in steradians: 2π π 1 average radiation intensity = U (θ, φ) sin θ dθ dφ = U0 (1-16) 4π 0 0 This is the radiated power divided by the area of a unit sphere. The radiation intensity U (θ, φ) separates into a sum of co- and cross-polarization components: U0 =

1 4π

2π 0

π 0

[UC (θ, φ) + U× (θ, φ)] sin θ dθ dφ

(1-17)

DIRECTIVITY ESTIMATES

11

Both co- and cross-polarization directivities can be deﬁned: directivityC =

UC,max U0

directivity× =

U×,max U0

(1-18)

Directivity can also be deﬁned for an arbitrary direction D(θ, φ) as radiation intensity divided by the average radiation intensity, but when the coordinate angles are not speciﬁed, we calculate directivity at Umax . 1-8 DIRECTIVITY ESTIMATES Because a ratio of radiation intensities is used to calculate directivity, the pattern may be referred to any convenient level. The most accurate estimate is based on measurements at equal angle increments over the entire radiation sphere. The average may be found from coarse measurements by using numerical integration, but the directivity measured is affected directly by whether the maximum is found. The directivity of antennas with well-behaved patterns can be estimated from one or two patterns. Either the integral over the pattern is approximated or the pattern is approximated with a function whose integral is found exactly. 1-8.1 Pencil Beam By estimating the integral, Kraus [4] devised a method for pencil beam patterns with its peak at θ = 0◦ . Given the half-power beamwidths of the principal plane patterns, the integral is approximately the product of the beamwidths. This idea comes from circuit theory, where the integral of a time pulse is approximately the pulse width (3-dB points) times the pulse peak: U0 = θ1 θ2 /4π, where θ1 and θ2 are the 3-dB beamwidths, in radians, of the principal plane patterns: directivity =

4π 41,253 (rad) = (deg) θ1 θ2 θ1 θ2

(1-19)

Example Estimate the directivity of an antenna with E- and H -plane (principal plane) pattern beamwidths of 24◦ and 36◦ . Directivity =

41,253 = 47.75 (16.8 dB) 24(36)

An analytical function, cos2N (θ/2), approximates a broad pattern centered on θ = 0◦ with a null at θ = 180◦ : U (θ ) = cos2N (θ/2)

or

E = cosN (θ/2)

The directivity of this pattern can be computed exactly. The characteristics of the approximation are related to the beamwidth at a speciﬁed level, Lvl(dB): beamwidth [Lvl(dB)] = 4 cos−1 (10−Lvl(dB)/20N ) N=

−Lvl(dB) 20 log[cos(beamwidthLvl(dB) /4)]

directivity = N + 1 (ratio)

(1-20a) (1-20b) (1-20c)

12

PROPERTIES OF ANTENNAS Directivity, dB

3-dB Beamwidth

SCALE 1-1 3-dB beamwidth and directivity relationship for cos2N (θ /2) pattern.

Directivity, dB

10-dB Beamwidth

SCALE 1-2 10-dB beamwidth and directivity relationship for cos2N (θ /2) pattern.

Scales 1-1 and 1-2, which give the relationship between beamwidth and directivity using Eq. (1-20), are useful for quick conversion between the two properties. You can use the two scales to estimate the 10-dB beamwidth given the 3-dB beamwidth. For example, an antenna with a 90◦ 3-dB beamwidth has a directivity of about 7.3 dB. You read from the lower scale that an antenna with 7.3-dB directivity has a 159.5◦ 10-dB beamwidth. Another simple way to determine the beamwidths at different pattern levels is the square-root factor approximation: BW[Lvl 2(dB)] Lvl 2(dB) = BW[Lvl 1(dB)] Lvl 1(dB) By this factor, beamwidth10 dB = 1.826 beamwidth3 dB ; an antenna with a 90◦ 3-dB beamwidth has a (1.826)90◦ = 164.3◦ 10-dB beamwidth. This pattern approximation requires equal principal plane beamwidths, but we use an elliptical approximation with unequal beamwidths: U (θ, φ) = cos2Ne (θ/2) cos2 φ + cos2Nh (θ/2) sin2 φ

(1-21)

where Ne and Nh are found from the principal plane beamwidths. We combine the directivities calculated in the principal planes by the simple formula directivity (ratio) =

2 · directivitye · directivityh directivitye + directivityh

(1-22)

Example Estimate the directivity of an antenna with E- and H -plane pattern beamwidths of 98◦ and 140◦ . From the scale we read a directivity of 6.6 dB in the E-plane and 4.37 dB in the H -plane. We convert these to ratios and apply Eq. (1-22): directivity (ratio) =

2(4.57)(2.74) = 3.426 or 4.57 + 2.74

10 log(3.426) = 5.35 dB

DIRECTIVITY ESTIMATES

13

Many analyses of paraboloidal reﬂectors use a feed pattern approximation limited to the front hemisphere with a zero pattern in the back hemisphere: U (θ ) = cos2N θ

or

E = cosN θ

◦

for θ ≤ π/2(90 )

The directivity of this pattern can be found exactly, and the characteristics of the approximation are beamwidth [Lvl(dB)] = 2 cos−1 (10−Lvl(dB)/20N ) −Lvl(dB) N= 20 log[cos(beamwidthLvl(dB) /2)] directivity = 2(2N + 1) (ratio)

(1-23a) (1-23b) (1-23c)

We use the elliptical model [Eq. (1-21)] with this approximate pattern and use Eq. (1-22) to estimate the directivity when the E- and H -plane beamwidths are different. 1-8.2 Butterﬂy or Omnidirectional Pattern Many antennas have nulls at θ = 0◦ with rotational symmetry about the z-axis (Figure 1-2). Neither of the directivity estimates above can be used with these patterns because they require the beam peak to be at θ = 0◦ . We generate this type of antenna pattern by using mode 2 log-periodic conical spirals, shaped reﬂectors, some higherorder-mode waveguide horns, biconical horns, and traveling-wave antennas. A formula similar to Kraus’s can be found if we assume that all the power is between the 3-dB beamwidth angles θ1 and θ2 : 1 θ2 cos θ1 − cos θ2 sin θ dθ = U0 = 2 θ1 2 Rotational symmetry eliminates integration over φ: directivity =

2 Umax = U0 cos θ1 − cos θ2

(1-24)

FIGURE 1-2 Omnidirectional antenna pattern with sidelobes scanned above the horizon.

14

PROPERTIES OF ANTENNAS Directivity, dB of Omnipattern at q = 90°

3-dB Beamwidth

SCALE 1-3 Relationship between 3-dB beamwidth of omnidirectional pattern and directivity.

Example A pattern with rotational symmetry has half-power points at 35◦ and 75◦ . Estimate the directivity. Directivity =

2 = 3.57 (5.5 dB) cos 35◦ − cos 75◦

If the pattern also has symmetry about the θ = 90◦ plane, the integral for the average radiation intensity has limits from 0 to π/2. Equation (1-24) reduces to directivity = 1/ cos θ1 . Example A rotationally symmetric pattern with a maximum at 90◦ has a 45◦ beamwidth. Estimate the directivity. θ1 = 90◦ − 45◦ /2 = 67.5◦ , so directivity =

1 = 2.61 cos 67.5◦

(4.2 dB)

The pattern can be approximated by the function U (θ ) = B sin2M (θ/2) cos2N (θ/2) but the directivity estimates found by integrating this function show only minor improvements over Eq. (1-24). Nevertheless, we can use the expression for analytical patterns. Given beam edges θL and θU at a level Lvl(dB), we ﬁnd the exponential factors. √ ln[cos(θU /2)] − ln[cos(θL /2)] AA = and T M2 = tan−1 AA ln[sin(θL /2)] − ln[sin(θU /2)] −|Lvl(dB)|/8.68589 N= AA{ln[sin(θL /2)] − ln(sin T M2 )} + ln(cos(θL /2)) − ln(cos T M2 ) M = AA(N ) A second pattern model of an omnidirectional pattern based on the pattern function with minor sidelobes and a beam peak at θ0 measured from the symmetry axis is sin[b(θ0 − θ )] b(θ0 − θ ) We estimate the directivity from the half-power beamwidth (HPBW) and the beam peak θ0 [5]: directivity(dB) = 10 log

101 (HPBW − 0.0027HPBW2 ) sin θ0

(1-25)

DIRECTIVITY ESTIMATES

15

Scan Factor, dB

Beam Direction, q

SCALE 1-4 pattern.

Additional directivity of omnidirectional pattern when scanned into conical

Scale 1-3 evaluates this formula for a beam at θ0 = 90◦ given HPBW, and Scale 1-4 gives the additional gain when the beam peak scans toward the axis. The directivity of butterﬂy patterns with unequal beamwidths in the principal planes cannot be estimated directly from the foregoing formulas. Similarly, some pencil beam patterns have large sidelobes which decrease the directivity and cannot be estimated accurately from Eq. (1-19). Both problems are solved by considering the directivity as an estimate of the average radiation intensity. Example A butterﬂy pattern peak is at 50◦ in both principal planes, but the beamwidths are 20◦ and 50◦ . Estimate the directivity. The 3-dB pattern points are given by: Cut 1 (40◦ and 60◦ ): U01 =

cos 40◦ − cos 60◦ = 0.133 2

U02 =

cos 25◦ − cos 75◦ = 0.324 2

Cut 2 (25◦ and 75◦ ):

Average the two pattern integral estimates: 0.133 + 0.324 = 0.228 2 1 Umax = = 4.38 (6.4 dB) directivity = U0 0.228 U0 =

Suppose that the beams are at different levels on the same pattern. For example, the lobe on the right of the ﬁrst pattern is the peak and the left lobe is reduced by 3 dB. The peaks of the second pattern are reduced by 1 dB. We can average on one pattern alone. Each lobe contributes Umax (cos θ1 − cos θ2 )/4 to the integral. The integral of the ﬁrst pattern is approximated by 0.266 + 0.266 × 10−3/10 = 0.100 4 The integral of the second pattern is reduced 1 dB from the peak. The average radiation intensity is found by averaging the two pattern averages: 0.100 + 0.324 × 10−0.1 = 0.178 2 1 = 5.602 (7.5 dB) directivity = U0 U0 =

16

PROPERTIES OF ANTENNAS

Pencil beam patterns with large sidelobes can be averaged in a similar manner: Up = 1/directivity. By using Eq. (1-19) and assuming equal beamwidths, we have Up = HPBW2 /41, 253, where Up is the portion of the integral due to the pencil beam and HPBW is the beamwidth in degrees. Example Consider a pencil beam antenna with pattern beamwidths of 50◦ and 70◦ in the principal planes. The second pattern has a sidelobe at θ = 60◦ down 5 dB from the peak and a 30◦ beamwidth below the 5 dB. What is the effect of the sidelobe on the directivity estimate? Without the sidelobe the directivity estimate is directivity =

41253 = 11.79 50(70)

(10.7 dB)

Consider each pattern separately: UP1 =

502 = 0.0606 41,253

UP2 =

702 = 0.1188 41,253

The sidelobe adds to the second integral: UPS2 =

(cos 45◦ − cos 75◦ )10−5/10 = 0.0354 4

Averaging the integrals of the parts gives us 0.1074: directivity =

1 = 9.31 (9.7 dB) U0

If there had been a sidelobe on each side, each would have added to the integral. Estimating integrals in this manner has limited value. Remember that these are only approximations. More accurate results can be obtained by digitizing the pattern and performing numerical integration on each pattern by using Eq. (1-16) or (1-17). 1-9 BEAM EFFICIENCY Radiometer system designs [6, p. 31–6] specify the antenna in terms of beam efﬁciency. For a pencil beam antenna with the boresight at θ = 0, the beam efﬁciency is the ratio (or percent) of the pattern power within a speciﬁed cone centered on the boresight to the total radiated power. In terms of the radiation intensity U ,

θ1

beam efﬁciency =

0

0

π

2π

U (θ, φ) sin θ dθ dφ (1-26)

0 2π

U (θ, φ) sin θ dθ dφ

0

where U includes both polarizations if necessary. Extended noise sources, such as radiometry targets, radiate noise into sidelobes of the antenna. Beam efﬁciency measures the probability of the detected target being located within the main beam (θ ≤ θ1 ).

INPUT-IMPEDANCE MISMATCH LOSS

17

Sometimes we can calculate directivity more easily than the pattern everywhere required by the denominator of Eq. (1-26): for example, a paraboloidal reﬂector. We use Eqs. (1-15) and (1-16) to calculate the denominator integral: π 2π 4πUmax U (θ, φ) sin θ dθ dφ = directivity 0 0 This reduces Eq. (1-26) to

θ1

2π

directivity beam efﬁciency =

0

U (θ, φ) sin θ dθ dφ

0

4πUmax

(1-27)

Equation (1-27) greatly reduces the pattern calculation requirements to compute beam efﬁciency when the directivity can be found without pattern evaluation over the entire radiation sphere. 1-10 INPUT-IMPEDANCE MISMATCH LOSS When we fail to match the impedance of an antenna to its input transmission line leading from the transmitter or to the receiver, the system degrades due to reﬂected power. The input impedance is measured with respect to some transmission line or source characteristic impedance. When the two are not the same, a voltage wave is reﬂected, ρV , where ρ is the voltage reﬂection coefﬁcient: ρ=

ZA − Z0 ZA + Z0

(1-28)

ZA is the antenna impedance and Z0 is the measurement characteristic impedance. On a transmission line the two traveling waves, incident and reﬂected, produce a standing wave: Vmax = (1 + |ρ|)Vi Vmin = (1 − |ρ|)Vi 1 + |ρ| Vmax = VSWR = Vmin 1 − |ρ|

(1-29) (1-30)

VSWR is the voltage standing-wave ratio. We use the magnitude of ρ, a complex phasor, since all the terms in Eq. (1-28) are complex numbers. The reﬂected power is given by Vi2 |ρ|2 /Z0 . The incident power is Vi2 /Z0 . The ratio of the reﬂected power to the incident power is |ρ|2 . It is the returned power ratio. Scale 1-5 gives the conversion between return loss and VSWR: return loss = −20 log |ρ| VSWR

Return Loss, dB

SCALE 1-5

Relationship between return loss and VSWR.

(1-31)

18

PROPERTIES OF ANTENNAS Reflected Power Loss, dB

Return Loss, dB

SCALE 1-6 Reﬂected power loss due to antenna impedance mismatch.

The power delivered to the antenna is the difference between the incident and the reﬂected power. Normalized, it is expressed as 1 − |ρ|2 or reﬂected power loss(dB) = 10 log(1 − |ρ|2 )

(1-32)

The source impedance to achieve maximum power transfer is the complex conjugate of the antenna impedance [7, p. 94]. Scale 1-6 computes the power loss due to antenna impedance mismatch. If we open-circuit the antenna terminals, the reﬂected voltage equals the incident voltage. The standing wave doubles the voltage along the transmission line compared to the voltage present when the antenna is loaded with a matched load. We consider the effective height of an antenna, the ratio of the open-circuit voltage to the input ﬁeld strength. The open-circuit voltage is twice that which appears across a matched load for a given received power. We can either think of this as a transmission line with a mismatch that doubled the incident voltage or as a Th´evenin equivalent circuit with an open-circuit voltage source that splits equally between the internal resistor and the load when it is matched to the internal resistor. Path loss analysis predicts the power delivered to a matched load. The mathematical Th´evenin equivalent circuit containing the internal resistor does not say that half the power received by the antenna is either absorbed or reradiated; it only predicts the circuit characteristics of the antenna load under all conditions. Possible impedance mismatch of the antenna requires that we derate the feed cables. The analysis above shows that the maximum voltage that occurs on the cable is twice that present when the cable impedance is matched to the antenna. We compute the maximum voltage given the VSWR using Eq. (1-29) for the maximum voltage: Vmax =

1-11

2Vi 2 VSWR(Vi ) = VSWR + 1 1 + 1/VSWR

(1-33)

POLARIZATION

The polarization of a wave is the direction of the electric ﬁeld. We handle all polarization problems by using vector operations on a two-dimensional space using the far-ﬁeld radial vector as the normal to the plane. This method is systematic and reduces chance of error. The spherical wave in the far ﬁeld has only θ and φ components of the electric ˆ Eθ and Eφ are phasor components in the direction of the unit ﬁeld: E = Eθ θˆ + Eφ φ. ˆ We can also express the direction of the electric ﬁeld in terms of vectors θˆ and φ. a plane wave propagating along the z-axis: E = Ex xˆ + Ey yˆ . The direction of propagation conﬁnes the electric ﬁeld to a plane. Polarization is concerned with methods

POLARIZATION

19

f (y)

wt

E

t q(x)

FIGURE 1-3

Polarization ellipse.

of describing this two-dimensional space. Both of the above are linear polarization expansions. We can rewrite them as ˆ E = Eθ (θˆ + ρˆL φ) E = Ex (ˆx + ρˆL yˆ )

Eφ Eθ Ey ρˆL = Ex ρˆL =

(1-34)

where ρˆL is the linear polarization ratio, a complex constant. If time is inserted into the expansions, and the tip of the electric ﬁeld traced in space over time, it appears as an ellipse with the electric ﬁeld rotating either clockwise (CW) or counter clockwise (CCW) (Figure 1-3). τ is the tilt of the polarization ellipse measured from the x-axis (φ = 0) and the angle of maximum response. The ratio of the maximum to minimum linearly polarized responses on the ellipse is the axial ratio. If ρˆL = e±j π/2 , the ellipse expands to a circle and gives the special case of circular polarization. The electric ﬁeld is constant in magnitude but rotates either CW (left hand) or CCW (right hand) at the rate ωt for propagation perpendicular to the page. 1-11.1 Circular Polarization Components The two circular polarizations also span the two-dimensional space of polarization. The right- and left-handed orthogonal unit vectors deﬁned in terms of linear components are ˆ = √1 (θˆ − j φ) ˆ R 2 1 ˆ Lˆ = √ (θˆ + j φ) 2

or or

ˆ = √1 (ˆx − j yˆ ) R 2 1 Lˆ = √ (ˆx + j yˆ ) 2

(1-35a) (1-35b)

The electric ﬁeld in the polarization plane can be expressed in terms of these new unit vectors: ˆ + ER R ˆ E = EL L

20

PROPERTIES OF ANTENNAS

When projecting a vector onto one of these unit vectors, it is necessary to use the complex conjugate in the scalar (dot) product: EL = E · Lˆ

∗

ˆ ER = E · R

∗

ˆ onto itself, we obtain When we project R ˆ ·R ˆ ∗ = 1 (θˆ − j φ) ˆ · (θˆ + j φ) ˆ = 1 (1 − j · j ) = 1 R 2 2 Similarly,

ˆ ∗ = 1 (θˆ + j φ) ˆ · (θˆ + j φ) ˆ = 1 (1 + j · j ) = 0 Lˆ · R 2 2

The right- and left-handed circular (RHC and LHC) components are orthonormal. A circular polarization ratio can be deﬁned from the equation ρˆc =

ˆ E = EL (Lˆ + ρˆc R)

ER = ρc ej δc EL

Let us look at a predominately left-handed circularly polarized wave when time and space combine to a phase of zero for EL . We draw the polarization as two circles (Figure 1-4). The circles rotate at the rate ωt in opposite directions (Figure 1-5), with the center of the right-handed circular polarization circle moving on the end of the vector of the left-handed circular polarization circle. We calculate the phase of the circular polarization ratio ρˆc from the complex ratio of the right- and left-handed circular components. Maximum and minimum electric ﬁelds occur when the circles

f (y)

ER

L +

E

R

E

EL

E

L −

E

R

ER

EL EL

q (x)

E

ER dc t

FIGURE 1-4 Polarization ellipse LHC and RHC components. (After J. S. Hollis, T. J. Lyons, and L. Clayton, Microwave Antenna Measurements, Scientiﬁc Atlanta, 1969, pp. 3–6. Adapted by permission.)

POLARIZATION

21

RHC f

LHC f Wave Propagating Out of Paper

E

wt q

E

wt

q

r^L = e −jp/2

r^L = e jp/2

FIGURE 1-5 Circular polarization components. (After J. S. Hollis, T. J. Lyons, and L. Clayton, Microwave Antenna Measurements, Scientiﬁc Atlanta, 1969, pp. 3–5. Adapted by permission.)

Axial Ratio, dB

Circular Cross-polarization, dB

SCALE 1-7 Circular cross-polarization/axial ratio.

alternately add and subtract as shown in Figure 1-4. Scale 1-7 shows the relationship between circular cross-polarization and axial ratio: √ √ Emax = (|EL | + |ER |) / 2 Emin = (|EL | − |ER |) / 2  1 + |ρˆc | |EL | + |ER | Emax   = LHC =  |EL | − |ER | 1 − |ρˆc | axial ratio = Emin (1-36) |ρˆc | + 1 |ER | + |EL | Emax   = RHC =  Emin |ER | − |EL | |ρˆc | − 1   |ρˆc | < 1 LHC 0 ≤ 1  < 1 RHC ρˆc axial ratio(dB) = 20 log

Emax Emin

The tilt angle of the polarization ellipse τ is one-half δc , the phase of ρˆc . Imagine time moving forward in Figure 1-5. When the LHC vector has rotated δc /2 CW, the RHC vector has rotated δc /2 CCW and the two align for a maximum. 1-11.2 Huygens Source Polarization When we project the currents induced on a paraboloidal reﬂector to an aperture plane, Huygens source radiation induces aligned currents that radiate zero cross-polarization

22

PROPERTIES OF ANTENNAS

in the principal planes. We separate feed antenna radiation into orthogonal Huygens sources for this case. To calculate the far-ﬁeld pattern of a paraboloid reﬂector, we can skip the step involving currents and integrate over the Huygens source ﬁelds in the aperture plane directly. We transform the measured ﬁelds of the feed into orthogonal Huygens sources by

Ec Eθf cos φ − sin φ = (1-37) sin φ cos φ Ex Eφf where Ec is the φ = 0 direction of polarization in the feed pattern and Ex is the φ = 90◦ polarization. This division corresponds to Ludwig’s third deﬁnition of crosspolarization [8]. The following matrix converts the Huygens source polarizations to the normal far-ﬁeld components of spherical coordinates:

Ec cos φ sin φ Eθ = (1-38) − sin φ cos φ Eφ Ex 1-11.3 Relations Between Bases In problems with antennas at arbitrary orientations, circularly polarized components have an advantage over linear components. When the coordinate system is rotated, both the amplitude and phase change for ρˆL , the linear polarization ratio, whereas the circular polarization ratio ρˆc magnitude is constant under rotations and only the phase changes. In other words, the ratio of the diameters of the circles (Figure 1-4) is constant. The circular components can be found from linear polarization components by projection. ˆ ∗ = √1 (Eθ θˆ + Eφ φ) ˆ ·R ˆ · (θˆ + j φ) ˆ ER = (Eθ θˆ + Eφ φ) 2 1 ER = √ (Eθ + j Eφ ) 2

(1-39)

Similarly, 1 EL = √ (Eθ − j Eφ ) 2 The linear polarizations can be found in terms of the circular components in the same manner: 1 j Eθ = √ (EL + ER ) Eφ = √ (EL − ER ) 2 2 These relations enable the conversion between polarizations. Good circularly polarized antennas over a wide bandwidth are difﬁcult to build, but good linearly polarized antennas are obtained easily. After we measure the phase and amplitude of Eθ and Eφ component phasors, we compute the circular components from Eq. (1-39), the axial ratio by using Eq. (1-36), and the polarization ellipse tilt τ from one-half the phase of ER /EL . We employ a leveled phase-locked source to record two patterns with orthogonal linear sources (or the same linear source is rotated between patterns). Afterward, we use the equations given above to convert polarization

23

POLARIZATION

to any desired polarization components. We calculate the maximum and minimum linear components by projecting the linear components into the rotated coordinate system of the polarization ellipse: Emax = Eθ cos τ + Eφ sin τ Emin = −Eθ sin τ + Eφ cos τ 1-11.4 Antenna Polarization Response The path loss formulas assume that the two antennas have matched polarizations. Polarization mismatch adds an extra loss. We determine polarization efﬁciency by applying the scalar (dot) product between normalized polarization vectors. An antenna transmitting in the z-direction has the linear components Ea = E1 (ˆx + ρˆL1 yˆ ) The incident wave on the antenna is given by Ei = E2 (ˆx + ρˆL2 yˆ ) where the wave is expressed in the coordinates of the source antenna. The z-axis of the source is in the direction opposite that of the antenna. It is necessary to rotate the coordinates of the receiving antenna wave. Rotating about the x-axis is equivalent to changing the sign of the tilt angle or taking the complex conjugate of Ea . The measurement antenna projects the incident wave polarization onto the antenna polarization. The antenna measures the incident ﬁeld, but we need to normalize the antenna polarization to a unit vector to calculate polarization efﬁciency: E2 · E∗1 =

∗ E2 E1∗ (1 + ρˆL2 ρˆL1 ) 2 1 + |ρˆL1 |

We normalize both the incident wave and antenna responses to determine loss due to polarization mismatch: xˆ + ρˆL2 yˆ Ei = ∗ |Ei | 1 + ρˆL2 ρˆL2

∗ xˆ + ρˆL1 E∗a yˆ = ∗ |Ea | 1 + ρˆL1 ρˆL1

The normalized voltage response is ∗ ρˆL2 Ei · E∗a 1 + ρˆL1 = ∗ ∗ |Ei ||Ea | 1 + ρˆL1 ρˆL1 1 + ρˆL2 ρˆL2

(1-40)

When we express it as a power response, we obtain the polarization efﬁciency : =

1 + |ρˆL1 |2 |ρˆL2 |2 + 2|ρˆL1 ||ρˆL2 | cos(δ1 − δ2 ) |Ei · E∗a |2 = |Ei |2 |Ea |2 (1 + |ρˆL1 |2 )(1 + |ρˆL2 |2 )

(1-41)

This is the loss due to polarization mismatch. Given that δ1 and δ2 are the phases of the polarization ratios of the antenna and the incident wave. As expressed in terms of linear polarization ratios, the formula is awkward because when the antenna is rotated to determine the peak response, both the amplitudes and phases change. A formula

24

PROPERTIES OF ANTENNAS

using circular polarization ratios would be more useful, because only phase changes under rotation. Two arbitrary polarizations are orthogonal ( = 0) only if |ρˆ1 | =

1 |ρˆ2 |

◦

and δ1 − δ2 = ±180

(1-42)

This can be expressed as vectors by using unit vectors: a1 · a∗2 = 0; a1 and a2 are the orthonormal generalized basis vectors for polarization. We can deﬁne polarization in terms of this basis with a polarization ratio ρ. By paralleling the analysis above for linear polarizations, we obtain the polarization efﬁciency for an arbitrary orthonormal polarization basis: =

1 + |ρˆ1 |2 |ρˆ2 |2 + 2|ρˆ1 ||ρˆ2 | cos(δ1 − δ2 ) (1 + |ρˆ1 |2 )(1 + |ρˆ2 |2 )

(1-43)

It has the same form as Eq. (1-41) derived for linear polarizations. We can use Eq. (1-43) with circular polarizations whose polarization ratio ρc magnitudes are constant with rotations of the antenna. The maximum and minimum polarization efﬁciencies occur when δ1 − δ2 equals 0◦ and 180◦ , respectively. The polarization efﬁciency becomes (1 ± |ρˆ1 ||ρˆ2 |)2 (1-44) max / min = (1 + |ρˆ1 |2 )(1 + |ρˆ2 |2 ) In all other vector pair bases for polarization, the magnitude of the polarization ratio ρ changes under rotations. Figure 1-6 expresses Eq. (1-44) as a nomograph. If we have ﬁxed installations, we can rotate one antenna until the maximum response is obtained and realize minimum polarization loss. In transmission between mobile antennas such as those mounted on missiles or satellites, the orientation cannot be controlled and the maximum polarization loss must be used in the link analysis. Circularly polarized antennas are used in these cases. Example A satellite telemetry antenna is RHC with an axial ratio of 7 dB. The ground station is RHC with a 1.5-dB axial ratio. Determine the polarization loss. Because the orientation of the satellite is unknown, we must use the maximum polarization loss. To ﬁnd it, use the RHC ends of the scales in Figure 1-6. Draw a line from 7 on the leftmost scale to 1.5 on the center scale. Read the loss on the scale between: 0.9 dB. The measured cross-polarization response of a linearly polarized antenna is the reciprocal of the axial ratio, the same absolute magnitude in decibels. Example Suppose that the linear cross-polarization responses of two antennas in a stationary link are given as 10 and 20 dB. Compute the minimum polarization loss. We rotate one of the antennas until the maximum response is found. The speciﬁcation of cross-polarization response does not state whether an antenna is predominately leftor right-handed circularly polarized. It must be one or the other. Suppose that the 20dB cross-polarization antenna is LHC. If the other antenna also is LHC, we use a line drawn from the lower portion of the center scale in Figure 1-6 to the rightmost LHC scale and read 0.2 dB of loss on the scale between the two. The second possibility is

POLARIZATION

25

FIGURE 1-6 Maximum and minimum polarization loss. (After A. C. Ludwig, A simple graph for determining polarization loss, Microwave Journal, vol. 19, no. 9, September 1976, p. 63.)

that the antenna could be predominately RHC. On drawing a line to the RHC (lower) scale, we read 0.7 dB on the center scale. When polarization is expressed in terms of linearly polarized components, it is ambiguous to give only magnitudes and no information of the circular polarization sense. 1-11.5 Phase Response of Rotating Antennas The polarization sense of an antenna can be determined from the phase slope of a rotating antenna. Before starting the phase measurement, determine that the setup is proper. Some older phase–amplitude receivers are ambiguous, depending on whether the local oscillator frequency was above or below the signal frequency. We use the convention that increased distance between antennas gives decreased phase. Move the antenna away from the source and observe decreasing phase or correct the setup. A rotating linearly polarized source ﬁeld is given by Es = E2 (cos α xˆ + sin α yˆ ) where α is clockwise rotation viewed from the direction of propagation (forward). A horizontally polarized linear antenna has the response Ea = E1 xˆ . It responds to the

26

PROPERTIES OF ANTENNAS

rotating linear source ﬁeld, E1 E2 cos α. The phase is constant under rotation until the null is passed and it ﬂips 180◦ through the null. An RHC polarized antenna has the response E1 (ˆx − j yˆ ). It responds to the rotating linear source ﬁeld, E1 E2 (cos α − j sin α) = E1 E2 e−j α The magnitude remains constant, but the phase decreases with rotation. Phase increases when the antenna is LHC. By observing the phase slope, the sense of the predominant polarization can be determined: RHC = negative phase slope; LHC = positive phase slope. It is easily remembered by considering the basis vectors of circular polarization: ˆ = √1 (ˆx − j yˆ ) R 2 In rotation from the x-axis to the y-axis, the phase decreases 90◦ . 1-11.6 Partial Gain If we measure the antenna gain to one polarization (e.g., RHC) and operate it in a link with an antenna also measured to one polarization, Eq. (1-44) fails to predict the response. Polarization efﬁciency assumes that the antenna gain was measured using a source ﬁeld with matched polarization. Gains referred to a single polarization are partial gains. If we align the two polarization ellipses of the two antennas, the response increases. Similarly, when the ellipses are crossed, the link suffers polarization loss. To obtain the full gain, we add the factor 10 log(1 + |ρ|2 )

(1-45)

to the partial gain, an expression valid using ρ for either circular or linear polarization. In terms of axial ratio A for circular polarization, the conversion is 2(1 + A2 ) 20 log 1+A When using measured partial gains for both antennas, the range of polarization efﬁciency is given by polarization efﬁciency = 20 log(1 ± ρ1 ρ2 )

(1-46)

We can convert Eq. (1-46) to expressions that use the axial ratio of the two antennas: 2(A1 A2 + 1) (A1 + 1)(A2 + 1) 2(A1 + A2 ) minimum polarization efﬁciency = 20 log (A1 + 1)(A2 + 1)

maximum polarization efﬁciency = 20 log

1-11.7 Measurement of Circular Polarization Using Amplitude Only The analyses given above assume that you can measure both amplitude and phase response of antennas, whereas in some cases only amplitude can be measured. If

VECTOR EFFECTIVE HEIGHT

27

you do not know the sense of circular polarization, it will be necessary to build two antennas that are identical except for their circular polarization sense. For example, you can build two identically sized counter-wound helical wire antennas. You determine polarization sense by using both sources and comparing measured levels. Once you establish the polarization sense, mount a linearly polarized measurement antenna with low cross-polarization. For a given pointing direction of the antenna under test, rotate the source antenna and record the maximum and minimum levels. The ratio of the maximum to the minimum is the axial ratio. To measure gain, rotate the measurement linearly polarized antenna to determine the peak response. Replace the antenna under test with a linearly polarized gain standard (horn) and perform a gain comparison measurement. Given the antenna axial ratio A, you adjust the linearly polarized gain by the correction factor: A+1 gain correction factor(dB) = 20 log √ 2A

(1-47)

We obtain the RHC and LHC response from 1 ER = √ (Emax + Emin ) 2

1 and EL = √ (Emax − Emin ) 2

assuming that the antenna is predominately RHC. 1-12 VECTOR EFFECTIVE HEIGHT The vector effective height relates the open-circuit voltage response of an antenna to the incident electric ﬁeld. Although we normally think of applying effective height to a line antenna, such as a transmitting tower, the concept can be applied to any antenna. For a transmitting tower, effective height is the physical height multiplied by the ratio of the average current to the peak current: VOC = Ei · h∗

(1-48)

The vector includes the polarization properties of the antenna. Remember from our discussion of antenna impedance mismatch that the open-circuit voltage√VOC is twice that across a matched load ZL for a given received power: VOC = 2 Prec ZL . The received power is the product of the incident power density S and the effective area of the antenna, Aeff . Gathering terms, we determine the open-circuit voltage from the incident ﬁeld strength E and a polarization efﬁciency : ZL Aeff VOC = 2E η We calculate polarization efﬁciency by using the scalar product between the normalized incident electric ﬁeld and the normalized vector effective height: =

|Ei ·h∗ |2 |Ei |2 |h|2

(1-49)

28

PROPERTIES OF ANTENNAS

Equation (1-49) is equivalent to Eq. (1-41) because both involve the scalar product between the incident wave and the receiving polarization, but the expressions have different normalizations. You can substitute vector effective height of the transmitting antenna for the incident wave in Eq. (1-49) and calculate polarization efﬁciency between two antennas. When an antenna rotates, we rotate h. We could describe polarization calculations in terms of vector effective height, which would parallel and repeat the discussion given in Section 1-11. We relate the magnitude of the effective height h to the effective area Aeff and the load impedance ZL : ZL Aeff (1-50) h=2 η The mutual impedance in the far ﬁeld between two antennas can be found from the vector effective heights of both antennas [9, p. 6–9]. Given the input current I1 to the ﬁrst antenna, we ﬁnd the open-circuit voltage of the second antenna: Z12 =

(V2 )OC j kηe−j kr h1 ·h∗2 = I1 4πr

(1-51)

When we substitute Eq. (1-50) into Eq. (1-51) and gather terms, we obtain a general expression for the normalized mutual impedance of an arbitrary pair of antennas given the gain of each in the direction of the other antenna as a function of spacing r: √ Z12 j G1 G2 −j kr h1 ·h∗2 = e (1-52) √ kr |h1 ||h2 | ZL1 ZL2 The magnitude of mutual impedance increases when the gain increases or the distance decreases. Of course, Eq. (1-52) is based on a far-ﬁeld equation and gives only an approximate answer, but it produces good results for dipoles spaced as close as 1λ. Figure 1-7 gives a plot of Eq. (1-52) for isotropic gain antennas with matched polarizations which shows the 1/R amplitude decrease with distance and that resistance and

Normalized Mutual Impedance

Reactance

Resistance

Spacing, l

FIGURE 1-7 Normalized mutual impedance (admittance) from the vector effective length for two antennas with 0 dB gain along the line between them.

MUTUAL COUPLING BETWEEN ANTENNAS

29

reactance curves are shifted out of phase. The cosine and sine factors of the complex exponential produce this effect. We multiply these curves by the product of the antenna gains, but the increased gain from larger antennas means that it is a greater distance to the far ﬁeld. When we bring two antennas close together, the currents on each antenna radiate and excite additional currents on the other that modify the result given by Eq. (1-52). But as we increase the distance, these induced current effects fade. Equivalent height analysis can be repeated using magnetic currents (e.g., used with microstrip patches), and Eqs. (1-51) and (1-52) become mutual admittance. Figure 1-7 is also valid for these antennas when we substitute normalized mutual admittance for normalized mutual impedance. For antennas with pattern nulls directed toward each other, the mutual impedance decreases at the rate 1/R 2 , due to the polarization of current direction h. 1-13 ANTENNA FACTOR The EMC community uses an antenna connected to a receiver such as a spectrum analyzer, a network analyzer, or an RF voltmeter to measure ﬁeld strength E. Most of the time these devices have a load resistor ZL that matches the antenna impedance. The incident ﬁeld strength Ei equals antenna factor AF times the received voltage Vrec . We relate this to the antenna effective height: AF =

Ei 2 = Vrec h

(1-53)

AF has units meter−1 but is often given as dB(m−1 ). Sometimes, antenna factor is referred to the open-circuit voltage and it would be one-half the value given by Eq. (1-53). We assume that the antenna is aligned with the electric ﬁeld; in other words, the antenna polarization is the electric ﬁeld component measured: 1 η 4π AF = = ZL Aeff λ ZL G This measurement may be corrupted by a poor impedance match to the receiver and any cable loss between the antenna and receiver that reduces the voltage and reduces the calculated ﬁeld strength. 1-14 MUTUAL COUPLING BETWEEN ANTENNAS The simplest approach for coupling between antennas is to start with a far-ﬁeld approximation. We can modify Eq. (1-8) for path loss and add the phase term for the ﬁnite distance to determine the S-parameter coupling: S21 =

e−j kr E1 ·E∗2 G1 G2 2kr |E1 ||E2 |

(1-54)

Equation (1-54) includes the polarization efﬁciency when the transmitted polarization does not match the receiving antenna polarization. We have an additional phase term

30

PROPERTIES OF ANTENNAS

because the signal travels from the radiation phase center along equivalent transmission lines to the terminals of each antenna. Equations (1-52) and (1-54) have the same accuracy except that Eq. (1-54) eliminates the need to solve the two-port circuit matrix equation for transmission loss. These formulas assume that antenna size is insigniﬁcant compared to the distance between the antennas, and each produces approximately uniform amplitude and phase ﬁelds over the second element. We can improve on Eq. (1-54) when we use the current distribution on one of the two antennas and calculate the near-ﬁeld ﬁelds radiated by the second antenna at the location of these currents. Since currents vary across the receiving antenna, we use vector current densities to include direction: Jr electric and Mr magnetic. Although magnetic current densities are ﬁctitious, they simplify the representation of some antennas. We compute coupling from reactance, an integral across these currents [see Eq. (2-34)]: j S21 = √ (Et ·Jr − Ht · Mr ) dV (1-55) 2 Pr Pt The input power to the transmitting antenna Pt produces ﬁelds Et and Ht . The power Pr into the receiving antenna excites the currents. The scalar product between the incident ﬁelds and the currents includes polarization efﬁciency. If we know the currents on the transmitting antennas, we calculate the near-ﬁeld pattern response from them at the location of the receiving antenna. Similar to many integrals, Eq. (1-55) is notional because we perform the integral operations only where currents exist. The currents could be on wire segments or surfaces. A practical implementation of Eq. (1-55) divides the currents into patches or line segments and performs the scalar products between the currents and ﬁelds on each patch and sums the result. A second form of the reactance [see Eq. (2-35)] involves an integral over a surface surrounding the receiving antenna. In this case each antenna radiates its ﬁeld to this surface, which requires near-ﬁeld pattern calculations for both. Equation (1-55) requires adding the phase length between the input ports and the currents, similar to using Eq. (1-54). When we use Eq. (1-55), we assume that radiation between the two antennas excites insigniﬁcant additional currents on each other. We improve the answer by using a few iterations of physical optics, which ﬁnds induced currents from incident ﬁelds (Chapter 2). We improve on Eq. (1-55) by performing a moment method calculation between the two antennas. This involves subdividing each antenna into small elements excited with simple assumed current densities. Notice the similarity between Eqs. (1-52) and (1-54) and realize that Eq. (1-55) is a near-ﬁeld version of Eq. (1-54). We use reactance to compute the mutual impedance Z21 between the small elements as well as their selfimpedance. For the moment method we calculate a mutual impedance matrix with a row and column for each small current element. We formulate a matrix equation using the mutual impedance matrix and an excitation vector to reduce coupling to a circuit problem. This method includes the additional currents excited on each antenna due to the radiation of the other.

1.15 ANTENNA NOISE TEMPERATURE [10] To a communication or radar system, an antenna contributes noise from two sources. The antenna receives noise power because it looks out on the sky and ground. The

ANTENNA NOISE TEMPERATURE

31

FIGURE 1-8 Antenna sky temperature. Noise temperature of an idealized antenna (lossless, no Earth-directed sidelobes) located at the Earth’s surface, as a function of frequency, for a number of beam elevation angles. Solid curves are for geometric-mean galactic temperature, sun noise 10 times quiet level, sun in unity-gain sidelobe, cool temperate-zone troposphere, 2.7 K cosmic blackbody radiation, zero ground noise. The upper dashed curve is for maximum galactic noise (center of galaxy, narrow-beam antenna). Sun noise 100 times quiet level, zero elevation, other factors the same as solid curves. The lower dashed curve is for minimum galactic noise, zero sun noise, 90◦ elevation angle. (The bump in the curves at about 500 MHz is due to the sun-noise characteristic. The curves for low elevation angles lie below those for high angles at frequencies below 400 MHz because of reduction of galactic noise by atmospheric absorption. The maxima at 22.2 and 60 GHz are due to the water-vapor and oxygen absorption resonance.) (From L. V. Blake, A guide to basic pulse-radar maximum-range calculation, Naval Research Laboratory Report 5868, December 1962.)

ground generates noise because it is about 290 K and a portion of the antenna pattern falls on it. Similarly, the sky adds noise dependent on the elevation angle and the operating frequency. Figure 1-8 gives the sky temperature versus frequency and elevation angle. The frequency range of lowest noise occurs in the middle of microwave frequencies of 1 to 12 GHz. The graphs show a large variation between the dashed curves, which occurs because of antenna direction and the pointing relative to the galactic center. In the middle of microwaves the sky noise temperatures are around 50 K, whereas near zenith the temperature is under 10 K. Near the horizon it rises because of the noise from oxygen and water vapor. The exact value must be determined for each application. As frequency decreases below 400 MHz, the sky temperature rises rapidly and becomes independent of antenna pointing. The curve continues the rapid rise at the same slope for lower frequencies. Low-frequency sky temperatures are often given as decibels relative to 290 K.

32

PROPERTIES OF ANTENNAS

An antenna receives this blackbody noise from the environment, but the value that affects the communication system depends both on the pattern shape and the direction of the main beam. We determine the antenna noise temperature by integrating the pattern times the environmental noise temperature distribution: Ta =

1 4π

2π 0

π

G(θ, φ)Ts (θ, φ) sin θ dθ dφ

(1-56)

0

where G(θ, φ) is the antenna gain pattern and Ts (θ, φ) is the angle-dependent blackbody radiation of the environment. Changing the antenna pointing changes Ta . Equation (1-56) is a weighted average of the environment noise temperature, usually referred to as the sky temperature. The second source of noise in the antenna is that of components that have both dissipative losses and reﬂection losses that generate noise. A receiving system needs to maximize the signal-to-noise ratio for given resources. System considerations, such as bit error rate, establish the required S/N ratio. We determine the noise power from the product N = k0 B n T e

(1-57)

where k0 is Boltzmann’s constant (1.38 × 10−23 W/K · Hz = −228.6 dB) and Bn is the receiver bandwidth (Hz). Te is the effective noise temperature (K). When referring noise temperature to other parts of the network, we increase or decrease it by the gain or loss, since it represents power and not a true temperature. Antenna gain is a measure of the signal level, since we can increase gain independent of the noise temperature, although the gain pattern is a factor by Eq. (1-56). The antenna conductor losses have an equivalent noise temperature: Te = (L − 1)Tp

(1-58)

where Tp is the antenna physical temperature and L is the loss (a ratio > 1). From a systems point of view, we include the transmission line run to the ﬁrst ampliﬁer or mixer of the receiver. We do not include the current distribution losses (aperture efﬁciencies) that reduce gain in Eq. (1-58) because they are a loss of potential antenna gain and not noise-generating losses (random electrons). The antenna–receiver chain includes mismatch losses, but these do not generate random electrons, only reﬂected waves, and have a noise temperature of zero. We include them in a cascaded devices noise analysis as an element with loss only. Noise characteristics of some receiver components are speciﬁed as the noise ﬁgure FN (ratio), and cascaded devices’ noise analysis can be analyzed using the noise ﬁgure, but we will use noise temperature. Convert the noise ﬁgures to noise temperature using TE = (FN − 1)T0

(1-59)

T0 is the standard reference temperature 290 K. We calculate noise temperature for the entire receiver chain of devices at a particular point normally at the input to the ﬁrst device. To calculate the S/N ratio we use the transmitter power, path loss (including antenna gain and polarization efﬁciency), and the gains (losses) of any devices for signal to the location in the receiver chain where

ANTENNA NOISE TEMPERATURE

33

noise temperature is being calculated. We characterize a given antenna by the ratio G/T , a measure independent of transmitter power and path loss, but including the receiver noise characteristics. Using the input of the ﬁrst device as the noise reference point, we calculate the input noise temperature from component noise temperatures and gains: T2 T3 T4 T = T1 + + + + ··· (1-60) G1 G1 G2 G1 G2 G3 Equation (1-60) merely states that noise temperatures are powers that decrease when we pass backward through a device with gain G. Each noise term is referred to the input of the device, and we pass backward to all previous devices and reduce noise temperature by 1/G. If we decided to locate the noise reference point at the input to the second device, the noise initially referred to the chain input would increase by the gain of the ﬁrst device. The system noise temperature becomes T(2) : T(2) = T1 G1 + T2 +

T3 T4 + + ··· G2 G2 G3

The signal also passes through the ﬁrst device and the new gain at the input to the second device becomes GG1 . The gain and the noise temperature change by the same factor G1 and produce a constant ratio. By extending these operations to any location in the receiver chain, we show that G/T is constant through the receiver device chain. It is easiest to illustrate G/T noise calculations with an example. A ground station has a 5-m-diameter paraboloid reﬂector with 60% aperture efﬁciency with the system operating at 2.2 GHz (λ = 0.136 m). We compute antenna directivity using the physical area and aperture efﬁciency:

π · Dia directivity = 0.60 λ

2

5π = 0.60 0.136

2 = 7972

(39 dB)

The reﬂector feed loss is 0.2 dB and it has a VSWR of 1.5 : 1. The cable between the feed and the ﬁrst ampliﬁer (LNA) has a 0.5-dB loss. These are elements under control of the antenna designer. We calculate the noise temperature of these by using Eq. (1-58) when we use a physical antenna temperature of 37.7◦ C (100◦ F) (310.8 K). Feed loss: T1 = (100.2/10 − 1)310.8 = 14.65 K Feed mismatch: T2 = 0 K Cable: T3 = (100.5/10 − 1)310.8 = 37.92 K The gains of these devices are G1 = 10−0.2/10 = 0.955 (feed loss), G2 = 10−0.18/10 = 0.959 (reﬂected power loss for 1.5 : 1 VSWR), and G3 = 10−0.5/10 = 0.891 (cable loss). The antenna sees the environment that generates noise due to blackbody radiation from the sky and ground. A typical value for the antenna pointed at 5◦ elevation is 50 K. This is not a physical temperature but represents an equivalent received power. Remember that the 60% aperture efﬁciency has no noise or loss contribution, because it only represents the loss of potential gain, since no random electrons are generated. We must consider the rest of the receiver chain when calculating the total input noise temperature. For this example we assume that the LNA has a noise ﬁgure of

34

PROPERTIES OF ANTENNAS

2 dB with 20 dB gain. The ﬁnal portion of the receiver includes the mixer and IF of the receiver, which we assume has a 10-dB noise ﬁgure. We use Eq. (1-59) to convert noise ﬁgure to noise temperature. LNA noise temperature T4 = 290(102/10 − 1) = 162.62 K Receiver noise temperature T5 = 290(1010/10 − 1) = 2610 K The 20-dB (100) LNA gain greatly reduces the effect of the 2610-K receiver. We calculate the contribution of each device to the input noise temperature by applying Eq. (1-60) to each device. We pass the noise temperature of the receiver through the four devices, and its temperature is reduced by the gain of each device: Te5 =

T5 2610 = 31.98 K = G1 G2 G3 G4 0.955(0.959)(0.891)(100)

The gain of the LNA greatly reduced the effective noise of the receiver at the antenna input. This operation shows that cascading noise temperature involves passing each device’s noise temperature through the gains of all preceding devices to the input and reducing it by the product of their gains. Similarly, we perform this operation on all the other noise temperatures. T4 169.62 = 207.86 K = G1 G2 G3 0.955(0.959)(0.891) T3 37.92 = 41.40 K Te3 = = G1 G2 0.955(0.959) T2 0 =0 = Te2 = G1 0.955 Te1 = T1 = 14.65 Te4 =

These operations illustrate that the cascaded devices’ noise temperature equation (1-60) is easily derived by considering the passage of noise temperature (power) through devices with gain to a common point where we can add the contributions. The sky temperature is not an input noise temperature but the noise power delivered at the ﬁctitious point called the antenna directivity, where gain = directivity. Since noise temperature represents power, we convert it to decibels and subtract it from directivity to compute G/T : G/T (dB) = 39 − 10 log(345.9) = 13.6 dB This G/T is a measure of the antenna and receiver combined performance when the antenna is pointed to 5◦ elevation. Changing the pointing direction affects only the sky temperature added directly to the ﬁnal result. We use G/T in the link budget of the communication system. We can supply a single value for the antenna gain and noise temperature at the output port connected to the receiver. Recognize that the ﬁrst three noise temperatures and the sky temperature are associated with the antenna. We moved the noise reference of each device to the input by dividing by the gain of the preceding devices. To move

COMMUNICATION LINK BUDGET AND RADAR RANGE

35

to the output of the antenna, we increase the noise temperature and the antenna gain by the product of the gain for the devices: T = (Tsky + Te1 + Te2 + Te3 )G1 G2 G3 = (50 + 14.65 + 0 + 37.92)10−0.88/10 = 83.7 K gain(dB) = directivity(dB) − 0.88 dB = 39 − 0.88 = 38.12 dB This reduces the antenna to a single component similar to the directivity and sky temperature that started our analysis. 1-16 COMMUNICATION LINK BUDGET AND RADAR RANGE We illustrate communication system design and path loss by considering a sample link budget example. The 5-m-diameter reﬂector is pointing at a satellite in an orbit 370 km above the Earth with a telemetry antenna radiating 10 W at 2.2 GHz. Since the antenna pattern has to cover the visible Earth, its performance is compromised. Considering the orbit geometry and antenna pointing is beyond the scope of this discussion. The range from a satellite at 370 km to a ground station pointing at 5◦ is 1720 km. The satellite antenna pointing angle from nadir is 70.3◦ , and a typical antenna for this application would have gain = −2 dBiC (RHC gain relative to an isotropic antenna) and an axial ratio of 6 dB. Assume that the ground station antenna has a 2-dB axial ratio. We apply the nomograph of Figure 1-6 to read the maximum polarization loss of 0.85 dB since we cannot control the orientation of the polarization ellipses. The link budget needs to show margin in the system, so we take worst-case numbers. When we apply Eq. (1-9) for path loss, we leave out the antenna gains and add them as separate terms in the link budget (Table 1-2): free-space path loss = 32.45 + 20 log[2200(1720)] = 164 dB The link budget shows a 4.4-dB margin, which says that the communication link will be closed. This link budget is only one possible accounting scheme of the system parameters. Everyone who writes out a link budget will separate the parameters differently. This budget shows typical elements. Radar systems have similar link budgets or detection budgets that consider S/N : S/N =

Prec PT GT (directivity)λ2 σ (EIRP)λ2 (G/T )σ = = KT B (4π)3 R 4 KT B (4π)3 R 4 KB

The radar has a required S/N value to enable it to process the information required, which leads to the maximum range equation:

(EIRP)λ2 (G/T )σ R= (4π)3 (S/N )req KB

1/4 (1-61)

Equation (1-61) clearly shows the role of the transmitter, EIRP; the receiver and antenna noise; G/T ; and the requirement for signal quality, S/Nreq , on the radar range for a given target size σ . Equation (1-61) applies to CW radar, whereas most radars use pulses. We increase radar performance by adding many pulses. We ignore the aspects of pulse train encoding

36

PROPERTIES OF ANTENNAS

TABLE 1-2 Link Budget Frequency Transmit power Transmit antenna gain EIRP (effective isotropic radiated power) Free-space path loss Polarization loss Atmospheric loss Rain loss Pointing loss Receive antenna directivity G/T Boltzmann’s constant Carrier/noise (C/N) ratio (ignores bandwidth) Bit rate: 8 Mb/s Eb /N0 (energy per bit/noise density) Implementation loss Eb /N0 required Margin

2.2 GHz 10 dBW −2 dB 8 dBW 164 dB 0.85 0.30 0.00 0.00 39 dB 13.6 dB 228.6 dB 85 dB 69 dB 16 dB 2 dB 9.6 dB 4.4 dB

Information only 10 log(10) Transmit power dBW + antenna gain dB Isotropic antenna path loss Maximum for uncontrolled orientation 5◦ elevation at 2.2 GHz Little loss at this frequency Location in receiver chain for G/T calculation From preceding section EIRP + G/T − path loss − polarization loss −atmospheric loss − rain loss + 228.6 10 log(bit rate) bandwidth Eb /N0 = C/N − 10 log(bit rate) Groups extra system losses For bit error rate (BER) = 10−5 in QPSK Eb /N0 − required Eb /N0 − implementation loss

that allow coherent addition. Radar range is determined by the total energy contained in the pulses summed. We replace EIRP with GT (energy) since PT × time = energy. It is the total energy that illuminates the target that determines the maximum detection range. Using antennas in radar leads to speaking of the radiated energy correct for pulsed systems, but when we do not integrate pulse shape times time, the antenna radiates power. To be correct we should call radiation that we integrate over angular space to ﬁnd power, “power density.” To say “energy radiated in the sidelobes” is poor physics unless it is a radar system, because it is power. 1-17

MULTIPATH

Multipath means that the ﬁeld intensity at a particular point is the sum of a number of waves that arrive from different directions or from different sources. It arises from signal transmission paths such as edge reﬂections from the mounting structure around an antenna and general reﬂections from objects near the antenna. Nearby reﬂections only seem to modify the antenna pattern, while reﬂections from additional objects cause rapid ripple with changing pattern angle. In Section 3-1 we discuss how to use the ripple angular rate and pattern distribution to locate its source. Multipath causes degraded system performance or measurement errors. Of course, multipath can improve performance as well. In fact, we add nearby objects, such as ground planes, to improve antenna performance. We specify pattern response in terms of the power response, but we add ﬁelds. An extra signal −20 dB relative to the main signal is 0.01 in power but 0.1 in ﬁeld strength

PROPAGATION OVER SOIL

37

Peak-to-Peak Ripple, dB

Interference Signal Level, dB

SCALE 1-8 Signal peak-to-peak amplitude ripple due to multipath signal. Maximum Phase Error (degrees)

Interference Signal Level, dB

SCALE 1-9 Peak phase error due to multipath signal.

(voltage). Since the extra signal can have any phase relative to the main signal, it can add or subtract. Given an extra signal MP(dB), the pattern ripple is ripple(dB) = 20 log

1 + 10MP/20 1 − 10MP/20

(1-62)

where MP(dB) has a negative sign. Scale 1-8 gives the relationship between peakto-peak amplitude ripple and the level of the multipath signal. Equation (1-62) is numerically the same as the relationship between return loss and 20 log(VSWR). The multipath signal can change the phase when summed with the main signal over a range given by maximum phase error = ± tan−1 (10MP/20 ) (1-63) Scale 1-9 calculates the peak phase error due to a multipath signal. 1-18 PROPAGATION OVER SOIL When we position antennas over soil and propagate the signal any signiﬁcant distance, it will reﬂect from soil or water and produce a large multipath signal. Soil is a conductive dielectric that reﬂects horizontally and vertically polarized signals differently. Typical ground constants are listed in Table 1-3. Given the grazing angle ψ measured between the reﬂected ray and ground, the voltage reﬂection coefﬁcients are sin ψ − εr − j x − cos2 ψ (εr − j x) sin ψ − εr − j x − cos2 ψ ρh = and ρv = sin ψ + εr − j x − cos2 ψ (εr − j x) sin ψ + εr − j x − cos2 ψ (1-64) where x = σ/ωε0 = 17, 975σ /frequency(MHz). Figure 1-9 gives the reﬂection coefﬁcient for the two polarizations versus grazing angle. Horizontal polarization reﬂects from soil about the same as a metal surface. Vertical polarization reﬂection produces a more interesting curve. The graph shows that the reﬂection is low over a region of grazing angles. The minimum reﬂection direction is called the Brewster angle. At this angle the reﬂected wave is absorbed

38

PROPERTIES OF ANTENNAS

TABLE 1-3

Typical Ground Constants

Surface Dry ground Average ground Wet ground Fresh water Seawater

Dielectric Constant

Conductivity (S)

4–7 15 25–30 81 81

0.001 0.005 0.020 0.010 5.0

Reflection, dB

Horizontal Polarization

Vertical Polarization

Grazing Angle

FIGURE 1-9

Average soil reﬂection for horizontal and vertical polarization.

into the soil. At high grazing angles ρh has a phase near 180◦ and ρv a phase of 0◦ . When the grazing angle decreases and becomes less than the Brewster angle, the vertical polarization reﬂection changes from 0◦ to 180◦ . Remember that for most general response nulls, the signal phase changes by 180◦ when passing through the transition. As the grazing angle approaches zero both reﬂection coefﬁcients approach −1 and multipath is independent of polarization. The electric ﬁeld at the receiving antenna is the sum of the direct wave plus the reﬂected wave, which traveled along a longer path: E = Ed [1 − exp(−j φ)] = Ed (1 − cos φ + j sin φ) We compute the magnitude |E| = |Ed | 1 + cos2 φ + sin2 φ − 2 cos φ = 2|Ed | sin(φ/2) for the small phase difference between the two equal-amplitude signals. The received power Prec is proportional to E 2 . The path loss for this multipath link is modiﬁed from the free-space equation: λ 2 hT hR 2 2 2πhT hR Prec = 4PT → PT GT GR GT GR sin (1-65) 4πd λd d2

MULTIPATH FADING

39

Equation (1-65) states that the power received is proportional to 1/d 4 and increases by h2 for either antenna. We can approximate the propagation over soil by a region for closely spaced antennas when the results consist of the free-space transmission with 1/d 2 average transmission with signiﬁcant variation due to multipath and a second region proportional to 1/d 4 with small multipath variations. The breakpoint between the two models occurs at a distance d = 4hT hR /λ. Experiments at mobile telephone frequencies showed that Eq. (1-65) overestimates the received power when the receiving antenna height is less than 30 m and a more correct model modiﬁes the exponent of hR [11, p. 38]: h2T hCR (1-66) d4 Below 10 m, C = 1 and the exponent varies linearly between 10 and 30 m: C = hR /20 + 12 . On a narrow-beam terrestrial propagation path, scattering from an object along a path an odd multiple of λ/2 produces a signal that reduces the main path signal. Given an obstacle at a distance h radial from the direct ray path and located dT from the transmitter and a distance dR from the receiving antenna, we determine the differential path length as λ nλdT dR h2 dT + dR =n (1-67) or clearance height h = = 2 dT dR 2 dT + dR Prec = PT GT GR

We call these Fresnel clearance zones of order n. The direct path should clear obstacles by at least one clearance zone distance h to prevent the scattered signal from having a negative impact on the communication link. The ﬁrst Fresnel zone touches ground when dT = 4hT hR /λ is the breakpoint distance between 1/d 2 and 1/d 4 propagation models. 1-19 MULTIPATH FADING Most mobile communication occurs when there is no direct path between the base station antennas and the mobile user. The signal reﬂects off many objects along the path between the two. This propagation follows a Rayleigh probability distribution about the mean signal level: r r2 R2 pr (r) = 2 exp − 2 prob[r < R] = PR (R) = 1 − exp − 2 α 2α 2α √ R is the signal level,√ α the value of the peak in the distribution, with mean = α π/2 and median RM = α 2 ln(2) = 1.1774α. The median signal level is found by ﬁtting measured data for various localities (town, small town, open country, etc.) into a prediction model. The signal will have large signal fades where the level drops rapidly. The Rayleigh model can be solved for the average distance between fades given the level. As a designer it is important to realize the magnitude of the problem [12, pp. 125–130]: 2(R/RM ) average distance between fades = λ √ 2π ln(2)(R/RM ) 2

(1-68)

40

PROPERTIES OF ANTENNAS Distance between Fades, l

Rayleigh Multipath Signal Fade, dB

SCALE 1-10

Average distance between fades and depth of fade for a Rayleigh multipath.

Length of Fade Region, l

Rayleigh Multipath Signal Fade, dB

SCALE 1-11

Average fade length and depth of fade for a Rayleigh multipath.

R is the fade level (ratio) and RM is the median signal level found from a propagation model. Scale 1-10 shows the relationship between the average distance between fades and the depth of fade for Rayleigh multipath. A mobile channel operating at 1.85 GHz (λ = 16.2 cm) has a 15-dB fade every 2.75λ which equals 44.5 cm, while 10-dB fades occur every 1.62λ = 26.25 cm. The communication system must overcome these fades. Fortunately, the deep fades occur over a short distance: 2(R/RM ) − 1 average length of fade = λ √ 2π ln(2)(R/RM ) 2

The signal fades and then recovers quickly for a moving user. Scale 1-11 shows the average fade length along a path given the depth of fade. For the 1.85-GHz channel the 15-dB fade occurs only over 0.06(16.2) = 0.97 cm, and the 10-dB fade length is 0.109(16.2) = 1.76 cm. The solution to mobile communication multipath fading is found either in increasing the link margin with higher gain base station antennas or the application of diversity techniques. We use multiple paths between the user and the base station so that while one path experiences a fade, the other one does not. Diversity has no effect on the median signal level, but it reduces the effects of the nulls due to the Rayleigh distribution propagation. REFERENCES 1. S. A. Schelkunoff and H. Friis, Antenna Theory and Practice, Wiley, New York, 1952. 2. R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961. 3. H. Friis, A note on a simple transmission formula, Proceedings of IRE, vol. 34, May 1946, pp. 254–256.

REFERENCES

41

4. J. D. Kraus, Antennas, McGraw-Hill, New York, 1950. 5. N. McDonald, Omnidirectional pattern directivity in the presence of minor lobes: revisited, IEEE Antennas and Propagation Magazine, vol. 41, no. 2, April 1999, pp. 63–65. 6. W. F. Croswell and M. C. Bailey, in R. C. Johnson and H. Jasik, eds., Antenna Engineering Handbook, McGraw-Hill, New York, 1984. 7. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits, McGraw-Hill, New York, 1948. 8. A. C. Ludwig, The deﬁnition of cross polarization, IEEE Transactions on Antennas and Propagation, vol. AP-21, no. 1, January 1973, pp. 116–119. 9. K. P. Park and C. T. Tai, Receiving antennas, Chapter 6 in Y. T. Lo and S. W. Lee, eds., Antenna Handbook, Van Nostrand Reinhold, New York, 1993. 10. L. V. Blake, Prediction of radar range, Chapter 2 in Radar Handbook, M. Skolnik, ed., McGraw-Hill, New York, 1970. 11. K. Fujimoto and J. R. James, Mobile Antenna Systems Handbook, 2nd ed., Artech House, Boston, 2001. 12. J. D. Parsons, The Mobile Radio Propagation Channel, Wiley, New York, 1992.

2 RADIATION STRUCTURES AND NUMERICAL METHODS

Antenna analysis, an important part of design, requires a compromise between extensive calculations and the fabrication and measurement of prototypes, which depends on your working environment. You should minimize cost, which means reducing the time from the start of a design to completion of a working model. In some cases you should not rush to build a prototype. For example, when designing large and expensive antennas, such as paraboloidal reﬂectors, the high fabrication cost justiﬁes the time required for analysis. Management will not let you proceed before knowing the design will work. You should develop a cost model for each design in which analysis is one factor. Analyses allow optimization of a design. You can design a number of antennas and adjust the dimensions until you ﬁnd the best one. Again, you should be considering the costs of your time. At some point the incremental improvements are not worth the extra time for further analyses. In any case, when you build the prototype, you can expect differences. You soon determine that you can achieve only limited knowledge about a design because fabrication and measurement errors mask the true response of the antenna. You are doing engineering, not a science project. Textbooks contain many analyses of ideal antennas, and this book is no exception. You need to consider the application and the ﬁnal antenna environment. The mounting structure has little effect on the pattern of a large antenna with narrow beamwidth because little radiation strikes it. The overall radiation characteristics of narrow- or wide-beam antennas depend signiﬁcantly on the shape of the vehicle and how the antenna is mounted. In later chapters we discuss how to use antenna mounting to improve performance, so you can take advantage of it. The size of the mounting structure limits the type of analysis used. In this chapter we discuss physical optics (PO) and geometric optics (GO) [geometric theory of diffraction (GTD)] for large structures. In physical optics we compute the current induced on the vehicle due to antenna radiation and include their radiation in the Modern Antenna Design, Second Edition, By Thomas A. Milligan Copyright  2005 John Wiley & Sons, Inc.

42

AUXILIARY VECTOR POTENTIALS

43

overall pattern. But the PO analysis cost rises rapidly as the number of small current patches increases for larger structures. PO analysis works well with large antennas, such as paraboloidal reﬂectors, that produce focused beams. Geometric optics uses ray optics techniques whose computation cost is independent of the size of the vehicle and whose accuracy improves as structure size increases. GO provides insight because we can visualize the combination of direct, reﬂected, and diffracted (GTD) rays to calculate the pattern, but it requires the solution of difﬁcult geometry problems. Smaller structures allow the use of multiple methods. For example, the moment method divides the surroundings into small patches and uses an expansion of the current in predetermined basis functions. This method uses integral equations of the boundary conditions to calculate a matrix equation involving coefﬁcients of the current expansion. Numerical methods invert the matrix to solve for the coefﬁcients, but it is a costly numerical operation and limits the size of the problem that can be handled to a few wavelengths. The ﬁnite-difference time-domain (FDTD) technique computes the ﬁelds on the structure in the time domain. This method handles moderate-sized structures and readily includes complex material properties such as biological features. FDTD divides the region into cubic cells and when excited by pulse feeding functions, it produces wide frequency bandwidth responses. Finite-element methods (FEMs) also divide the problem into cubic cells, but the analysis is performed in the frequency domain. FEM analysis must be repeated at every frequency of interest. FDTD and FEMs require a program to divide the structure into a mesh before starting the solution. Both methods calculate currents on a boundary surface by using the equivalence theorem with the incident ﬁelds and then calculate the far-ﬁeld radiation pattern from these boundary currents. Most methods start by assuming a current distribution on the antenna or, equivalently, a distribution of ﬁelds on an aperture. The ﬁelds on the aperture can be reduced to a current distribution. The moment method uses a summation of assumed basis function currents and solves for the coefﬁcients of the expansion, but it, too, starts with assumed currents over small regions. You will discover that the radiation pattern can be found with greater accuracy than the input impedance. For antennas constructed from wires, the moment method computes the input current for a given excitation voltage and we calculate impedance from the ratio. Interaction of an antenna with the currents induced on a structure has little effect on impedance for narrow-beam antennas. Even for wide-beam antennas, such as dipoles, the structure effect on impedance can be found by using source mutual coupling with its images. In the end, antenna impedance should be measured when mounted in the ﬁnal conﬁguration. An antenna has both a radiation pattern bandwidth and an impedance bandwidth, but you must give the pattern primary consideration. Too many designs concentrate on the wideband impedance characteristics of an antenna when, in fact, the antenna pattern has changed over the frequency range of the impedance bandwidth. Your primary task should be to design for the radiation pattern desired. In Chapter 1 we detailed the system aspects of impedance mismatch (Section 1-10), and you may determine the overall system impact of small impedance mismatch. 2-1 AUXILIARY VECTOR POTENTIALS We do not use vector potentials in design. It seems as though they would be useful, but only a few simple antennas ﬁt their direct use. You cannot measure them

44

RADIATION STRUCTURES AND NUMERICAL METHODS

because they are not physical entities, so they seem artiﬁcial. Physical optics (PO) calculates the radiation directly from currents using dyadic Green’s functions but uses long expressions. Nevertheless, many analysis techniques ﬁnd them more efﬁcient than PO expressions and you should be aware of them. We illustrate their use with a couple of simple antennas. We use vector potentials to introduce a few antenna concepts. In the ﬁrst example we apply the magnetic vector potential to calculate the radiation from a short-length current element (dipole) and show how to obtain the pattern. Integration of the radiation pattern power density (Section 1-2) determines the total power radiated. Because we know the input current and the total radiated power, the ratio of the power to the input current squared gives the radiation resistance. We combine the low radiation resistance with the material resistance to compute the antenna efﬁciency. Electric vector potentials used with ﬁctitious magnetic currents illustrate analysis by duality. We apply this to the analysis of a small loop and show that it has the same pattern as that of a small dipole. 2-1.1 Radiation from Electric Currents Normal electron currents radiate when time varying. The simplest example is a ﬁlamentary current on wire, but we include surface and volumetric current densities as well. We analyze them by using the magnetic vector potential. Far-ﬁeld electric ﬁelds are proportional to the magnetic vector potential A: E = −j ωA

(2-1)

We determine the magnetic ﬁeld from |E| = η|H|

(2-2)

and realizing the cross product of the electric ﬁeld with the magnetic ﬁeld points in the direction of power ﬂow, the Poynting vector. Since the electric ﬁeld direction deﬁnes polarization, we usually ignore the magnetic ﬁeld. We derive the magnetic vector potential from a retarded volume integral over the current density J: A=µ

J(r )e−j k|r−r | dV 4π|r − r |

(2-3)

where r is the ﬁeld measurement point radius vector, r the source-point radius vector, µ the permeability (4π × 10−7 A/m in free space), and k, the wave number, is 2π/λ. As written, Eq. (2-3) calculates the potential A everywhere: near and far ﬁeld. The vector potential can be written in terms of a free-space Green’s function: e−j kR where R = |r − r | 4πR A=µ g(R)J(r ) dV

g(R) =

(2-4)

Radiation Approximation When we are interested only in the far-ﬁeld response of an antenna, we can simplify the integral [Eq. (2-3)]. An antenna must be large in terms

AUXILIARY VECTOR POTENTIALS

45

of wavelengths before it can radiate efﬁciently with gain, but at great distances it still appears as a point source. Consider the radiation from two different parts of an antenna. Far away from the antenna, the ratio of the two distances to the different parts will be nearly 1. The phase shift from each part will go through many cycles before reaching the observation point, and when adding the response from each part, we need only the difference in phase shift. In the radiation approximation we pick a reference point on the antenna and use the distance from that point to the far-ﬁeld observation point for amplitudes, 1/R, for all parts of the antenna. The direction of radiation deﬁnes a plane through the reference point. This plane is deﬁned by the radius normal vector, given in rectangular coordinates by rˆ = sin θ cos φ xˆ + sin θ sin φ yˆ + cos θ zˆ We compute the phase difference to the far-ﬁeld point by dropping a normal to the reference plane from each point on the antenna. This distance multiplied by k, the propagation constant, is the phase difference. Given a point on the antenna r , the phase difference is kr · rˆ . When we substitute these ideas into Eq. (2-3), the equation becomes e−j kr µ J ej kr ·ˆr dV A= (2-5) 4πr In rectangular coordinates kr · rˆ becomes k(x sin θ cos φ + y sin θ sin φ + z cos θ ) We can combine k and rˆ to form a k-space vector: k = kˆr = k sin θ cos φ xˆ + k sin θ sin φ yˆ + k cos θ zˆ and the phase constant becomes k · r . Currents in ﬁlaments (wires) simplify Eq. (2-5) to a single line integral. Magnetic vector potentials and electric ﬁelds are in the same directions as the wires that limit the directions of current. For example, ﬁlamentary current along the z-axis produces z-directed electric ﬁelds. Spherical waves (far ﬁeld) have only θˆ and φˆ components found from the projection of Ez onto those axes. Filamentary currents on the z-axis produce only z-directed electric ﬁelds with a null from θˆ · zˆ = − sin θ at θ = 0. In turn, x- or y-directed currents produce electric ﬁelds depending on the scalar products (projections) of the xˆ and yˆ unit vectors onto the θˆ and φˆ vectors in the far ﬁeld: θˆ · xˆ = cos θ cos φ θˆ · yˆ = cos θ sin φ

φˆ · xˆ = − sin φ φˆ · yˆ = cos φ

By examining antenna structure you can discover some of its characteristics without calculations. Without knowing the exact pattern, we estimate the polarization of the waves by examining the directions of the wires that limit the current density. Consider various axes or planes of symmetry on an antenna: for example, a center-fed wire along the z-axis. If we rotate it about the z-axis, the problem remains the same, which means that all conical polar patterns (constant θ ) must be circles; in other words, all great

46

RADIATION STRUCTURES AND NUMERICAL METHODS

circle patterns must be the same. An antenna with the same structure above and below the x –y plane radiates the same pattern above and below the x –y plane. Always look for axes and planes of symmetry to simplify the problem. We can extend the magnetic vector potential [Eq. (2-1)] to determine near ﬁelds: E = −j ωA + H=

∇(∇ · A) j ωεµ

1 ∇ ×A µ

(2-6)

The electric ﬁeld separates into far- and near-ﬁeld terms, but the equation for the magnetic ﬁeld, the deﬁning equation of the potential, does not separate. If we substitute the free-space Green’s function from Eq. (2-4) into Eq. (2-6), expand, and gather terms, we can determine the ﬁelds directly from the electric currents and eliminate the use of a vector potential. 1 j j ηk 2 − + 3 3 J(r ) − E(r) = 4π kR k 2 R 2 k R V

3 j 3j + 2 2− 3 3 e−j kR dV kR k R k R k2 1 j ˆ H(r) = + 2 J(r ) × R e−j kR dV 4π kR k R ˆ R ˆ +[J(r ) · R]

V

ˆ = r−r = r−r R |r − r | R

since

(2-7) (2-8)

R = |r − r |

Terms with 1/R dependence are the far-ﬁeld terms. The radiative near-ﬁeld terms have 1/R 2 dependence and near-ﬁeld terms have 1/R 3 dependence. The impedance of free space, η, is 376.7 . We can rearrange Eqs. (2-7) and (2-8) so that they become the integral of the dot product of the current density J with dyadic Green’s functions [1]. It is only a notation difference that leads to a logic expression. Except for a few examples given below, we leave the use of these expressions to numerical methods when designing antennas. Example Use the magnetic vector potential to derive the far ﬁeld of a short-length current element. Assume a constant current on the wire. The current density is I lδ(r ), where δ(r ) is the Dirac delta distribution and l is the length over which the far-ﬁeld phase is constant. The integral in Eq. (2-4) easily reduces to µI le−j kr 4πr The current element is so short that the phase distances from all parts of the element are considered to be equal; e−j kr is the retarded potential phase term. The electric ﬁeld is found from Az using Eq. (2-1): I l −j kr Ez = −j ωµ e 4πr I l −j kr Eθ = Ez zˆ · θˆ = j ωµ sin θ e 4πr Az =

AUXILIARY VECTOR POTENTIALS

47

√ √ √ µ µ, and divide and multiply by ε: √ j I l2πf µε µ −j kr sin θ Eθ = e 4πr ε

Evaluate ω as 2πf , split µ in

The following terms can be recognized as l f = c λ

1 c= √ µε

η=

µ ε

The far-ﬁeld electric ﬁeld becomes j I lη −j kr sin θ e 2λr The magnetic ﬁeld is found from the electric ﬁeld using Eq. (2-2): Eθ =

Hφ =

j I l −j kr Eθ = e sin θ η 2λr

The term j can be evaluated as ej π/2 , a phase shift term. The power density Sr is Sr = Eθ Hφ∗ =

|I |2 l 2 η sin2 θ 4λ2 r 2

The normalized power pattern is equal to sin2 θ . Figure 2-1 gives the polar pattern of this antenna as a dashed plot. The dashed circle is the −3-dB pattern level. We measure the angular separation between the 3-dB points to determine the beamwidth (half-power beamwidth). For comparison, Figure 2-1 shows the pattern of a half-wavelength-long dipole as a solid curve. At a length about 5% shorter than a half wavelength, the reactive component of the impedance vanishes. The ﬁgure illustrates that a short dipole has about the same pattern as a long-resonant-length (reactance equals zero) dipole. We determine directivity (Section 1-7) by calculating the average radiation intensity, often normalized to the peak of the power pattern: π/2 2 Uavg = sin2 θ sin θ dθ = 3 0 Umax = 1 Umax directivity = = 1.5 (1.76 dB) Uavg The resonant-length dipole (≈ λ/2) has a directivity of 2.15 dB or only 0.39 dB more than that of the very short dipole. The total power radiated by the antenna is found by integrating the Poynting vector magnitude over a sphere: 2π π Pr = Sr r 2 sin θ dθ dφ

0

0 π

|I |l = 2λ 0 0 2 2π |I |l η = 3 2λ 2π

2 η sin θ dθ dφ

48

RADIATION STRUCTURES AND NUMERICAL METHODS

l/2 Dipole

Short Dipole

FIGURE 2-1 Pattern of a short current element and small loop (dashed curve) compared to a λ/2 -long dipole (solid curve) located along the 0 to 180◦ axis.

We represent the radiated power as a radiation resistance at the input of the antenna: 2 l Pr 2π η = RR = |I |2 3 λ While a short dipole with a length λ/20 has a radiation resistance of about 2 , a resonant-length dipole has about a 50- radiation resistance and is more efﬁcient because the relative material resistance is low. The input resistance of the antenna is the sum of the radiation resistance and the resistance due to material losses: Pin = (RR + RL )|I |2 The gain of an antenna is the ratio of the peak radiation intensity to the input power averaged over the radiation sphere: gain =

Sr,peak r 2 Umax = Pin Pin 4π 4π

By using the idea of radiation resistance, we rewrite this as gain =

4πUmax (RR + RL )|I |2

AUXILIARY VECTOR POTENTIALS

49

Efﬁciency is the ratio of radiated power to input power: ηe =

Pr Rr |I |2 Rr = = Pin (Rr + RL )|I |2 Rr + RL

Instead of integrating the pattern to calculate the total power radiated, we sometimes compute the input power of the antenna from currents induced on the antenna elements by given voltage sources on various terminals of the antenna in analysis: Pin = Re(V1 I1∗ ) + Re(V2 I2∗ ) + · · · + Re(VN IN∗ ) The gain can be found from gain =

Sr (θ, φ)r 2 U (θ, φ) = Pin Pin 4π 4π

This method is considerably easier than integrating the radiation intensity to compute directivity. By integrating the pattern, we found only the input resistance of the short antenna, not the reactive component. A short antenna has a large capacitive reactance term that limits the impedance bandwidth when combined with a match network. The short antenna has a large pattern bandwidth but a narrow impedance bandwidth. Of course, an active network could be designed to impedance-match the antenna at any frequency, but the instantaneous bandwidth is narrow. The moment method of analysis gives us the currents for given input voltages and calculates the complete input impedance. 2-1.2 Radiation from Magnetic Currents Magnetic currents are ﬁctitious, but they enable slot radiation to be solved by the same methods as electric currents on dipoles by using duality. Slot radiation could be calculated from the surface currents around it, but it is easier to use magnetic currents to replace the electric ﬁeld in the slot. Magnetic currents along the long axis of slots in ground planes replace the electric ﬁelds across the slots by application of the equivalence theorem. Similarly, current loops can be replaced by magnetic dipole elements to calculate radiation. We use the electric vector potential F with magnetic currents. The far-ﬁeld magnetic ﬁeld is proportional to the electric vector potential: H = −j ωF

(2-9)

We determine the magnitude of the electric ﬁeld from Eq. (2-2); it is perpendicular to H. The electric vector potential is found from a retarded volume integral over the magnetic current density M. Applying the radiation approximation, it is e−j kr F= (2-10) ε M ej k·r dV 4πr where ε is the permittivity (8.854 × 10−12 F/m in free space). Equation (2-9) is the dual of Eq. (2-1), and Eq. (2-10) is the dual of Eq. (2-5). The dual of Eq. (2-3) is valid in both the near- and far-ﬁeld regions.

50

RADIATION STRUCTURES AND NUMERICAL METHODS

The magnetic currents in a slot are perpendicular to the slot electric ﬁelds: M = E × n, ˆ where nˆ is the normal to the plane with the slot. The ﬁlamentary currents of thin slots reduce Eq. (2-10) to a line integral, and magnetic current direction limits the direction of the electric vector potential and the magnetic ﬁeld. Since the electric ﬁeld (far ﬁeld) is orthogonal to the magnetic ﬁeld, the electric ﬁeld is in the same direction as the ﬁeld across the slots. We use the direction of the electric ﬁeld across the slots to estimate the polarization of the far ﬁeld. As with ﬁlamentary electric currents, the far ﬁeld is zero along the axis of the magnetic current. The electric vector potential can also be used to derive the near ﬁeld: H = −j ωF +

∇(∇ · F) j ωµε

1 E = − ∇ ×F ε The magnetic ﬁeld separates into near- and far-ﬁeld terms in the electric vector potential; the electric ﬁeld does not. We can determine the radiated ﬁelds directly in terms of the magnetic currents and avoid using the vector potential: E(r) = −

k2 4π

ˆ M(r ) × R

V

1 j + 2 e−j kR dV kR k R

1 j j − + 3 3 M(r ) − kR k 2 R 2 k R V j 3j 3 ˆ R ˆ e−j kR dV +[M(r ) · R] + 2 2− 3 3 kR k R k R

k H(r) = 4πη 2

(2-11)

(2-12)

Equations (2-11) and (2-12) can be rearranged to ﬁnd the dyadic Green’s functions for magnetic currents. These differ from the dyadic Green’s functions for electric currents by only constants. Example Derive the ﬁelds radiated from a small constant-current loop. We could use the magnetic vector potential and calculate over the currents in the wire but must account for changing current direction around the loop. Place the loop in the x –y plane. The electric ﬁeld radiated by the loop is in the φ direction because the currents in the loop can only be in the φˆ direction. When solving the integral for the magnetic vector potential, note that the direction of the current on the loop, φˆ at a ˆ unit vector. The integral general point is not in the same direction as the ﬁeld point, φ, must be solved with a constant vector direction, one component at a time. Although the magnetic vector potential can be computed, it is easier to replace the current loop with an incremental magnetic current element. The equivalent magnetic current element is Im l = j ωµI A where A is the area of the loop. The magnetic current density is M = Im lδ(r )ˆz = j ωµI Aδ(r )ˆz

APERTURES: HUYGENS SOURCE APPROXIMATION

51

The electric vector potential is found using Eq. (2-10): Fz =

j ωµεI A −j kr e 4πr

The magnetic ﬁeld is found from this electric vector potential using Eq. (2-9): Hz = −j ωFz =

ω2 µεI A −j kr e 4πr

We calculate Hθ by projection: ω2 µεI A −j kr e sin θ Hθ = Hz zˆ · θˆ = − 4πr Eφ and Hθ are related in the far ﬁeld because the wave propagates in the r direction: Eφ = −ηHθ =

ω2 µεI Aη −j kr e sin θ 4πr

The small current loop and small current element have the same pattern shape, sin θ , but opposite polarizations. The directivity is 1.5 (1.76 dB). Figure 2-1 uses a dashed curve to plot the response of the small loop, while the solid curve gives the pattern of a half-wavelength slot that radiates on both sides of the ground sheet. 2-2 APERTURES: HUYGENS SOURCE APPROXIMATION Many antennas, such as horns or paraboloid reﬂectors, can be analyzed simply as apertures. We replace the incident ﬁelds in the aperture with a combination of equivalent electric and magnetic currents. We calculate radiation as a superposition of each source by using the vector potentials. Most of the time, we assume that the incident ﬁeld is a propagating free-space wave whose electric and magnetic ﬁelds are proportional to one another. This gives us the Huygens source approximation and allows the use of integrals over the electric ﬁeld in the aperture. Each point in the aperture is considered to be a source of radiation. The far ﬁeld is given by a Fourier transform of the aperture ﬁeld: f(kx , ky ) = Eej k·r ds (2-13) S

This uses the vector propagation constant k = kx xˆ + ky yˆ + kz zˆ kx = k sin θ cos φ

ky = k sin θ sin φ

kz = k cos θ

where f(kx , ky ) is the pattern in k-space. We multiply the Fourier transform far ﬁeld by the pattern of the Huygens source: j e−j kr (1 + cos θ ) 2λr

(2-14)

When apertures are large, we can ignore this pattern factor. In Eq. (2-13), f(kx , ky ) is a vector in the same direction as the electric ﬁeld in the aperture. Each component

52

RADIATION STRUCTURES AND NUMERICAL METHODS

is transformed separately. The far-ﬁeld components Eθ and Eφ are found by projection (scalar products) from f(kx , ky ) times the pattern factor of the Huygens source [Eq. (2-14)]. If we have a rectangular aperture in which the electric ﬁeld is expressed as a product of functions of x and y only, the integral reduces to the product of two single integrals along each coordinate. The Fourier transform relationships provide insight into pattern shape along the two axes. Large apertures radiate patterns with small beamwidths. An antenna with long and short axes has a narrow-beamwidth pattern in the plane containing the long dimension and a wide beamwidth in the plane containing the short dimension. This is the same as the time and frequency dual normally associated with the Fourier transform. We draw on our familiarity with signal processing to help us visualize the relationship between aperture distributions and patterns. Large apertures give small beamwidths, just as long time pulses relate to low-frequency bandwidths in normal time–frequency transforms. The sidelobes of the pattern correspond to the frequency harmonics of an equivalent time waveform under the Fourier transform and rapid transitions in the time response lead to high levels of harmonics in the frequency domain. Rapid amplitude transitions in the aperture plane produce high sidelobes (harmonics) in the far-ﬁeld response (Fourier transform). Step transitions on the aperture edges produce high sidelobes, while tapering the edge reduces sidelobes and we relate the sidelobe envelope of peaks to the derivative of the distributions at the edges. To produce equal-level sidelobes, we need Dirac delta functions in the aperture that transform to a constant level in the pattern domain. Another example is periodic aperture errors that produce single high sidelobes. When we discuss aperture distribution synthesis, we see that the aperture extent in wavelengths limits our ability to control the pattern. A uniform amplitude and phase aperture distribution produces the maximum aperture efﬁciency and gain that we determine from the following argument. An aperture collects power from a passing electromagnetic wave and maximum collectible power occurs at its peak amplitude response. If the amplitude response somewhere else in the aperture is reduced from the maximum, that portion will collect less power. The amplitude response can be reduced only by adding loss or reﬂecting power in reradiation. The antenna with the highest aperture efﬁciency reﬂects the least amount of power when illuminated by a plane wave. Similarly, if the phase shift from the collecting aperture to the antenna connector is different for different parts of the aperture, the voltages from the various parts will not add in phase. Gain is directly proportional to aperture efﬁciency [Eq. (1-10)]. Therefore, a uniform amplitude and phase aperture distribution has maximum gain. All this assumes that the input match on various aperture distribution antennas is the same. For example, consider the pattern of a uniform aperture distribution in a rectangular aperture a × b. We use the Fourier transform and ignore the polarization of the electric ﬁeld in the aperture. (This assumes that the ﬁeld has a constant polarization or direction.) b/2 a/2 ej k·r dx dy f (kx , ky ) = E0 = E0

−b/2 b/2 −b/2

−a/2 a/2 −a/2

ej kx x ej ky y dx dy

APERTURES: HUYGENS SOURCE APPROXIMATION

53

We separate the integral into a product of two integrals each with the form a/2 ej kx a/2 − e−j kx a/2 a sin(kx a/2) ej kx x dx = = j k kx a/2 x −a/2 On combining the two similar integrals, we have f (kx , ky ) =

ab sin(kx a/2) sin(ky b/2) kx a/2 ky b/2

where kx = k sin θ cos φ, ky = k sin θ sin φ, and kz = k cos θ and k = 2π/λ. The pattern in both planes is given by a k-space function, sin u/u. Figure 2-2 plots this pattern function as a solid curve using kx -space [(ka/2) sin θ ] as the abscissa to produce a universal curve independent of aperture size a. The half-power points occur when sin u 1 =√ u 2

or

u = 1.39156

When we substitute for u, we have in the principal planes πa sin θ = 1.39156 λ By solving for θ , we compute the half-power beamwidth (HPBW): HPBW = 2 sin−1

0.4429λ a

By using the approximation u = sin u for small angles, the half-power beamwidth can be estimated as ◦λ HPBW = 50.76 a

Amplitude, dB

Cosine Uniform

ka/2 sin q

FIGURE 2-2 Universal k-space pattern for the radiation from uniform (solid curve) and cosine (dashed curve) aperture distributions.

54

RADIATION STRUCTURES AND NUMERICAL METHODS

Note that we have ignored the (1 + cos θ )/2 pattern of the Huygens source, which reduces the beamwidth for radiation from small apertures. We discover on Figure 2-2 the pattern nulls occur at integer multiples of π and the ﬁrst sidelobe amplitude is 13.2 dB below the peak. The gain of a uniform amplitude and phase aperture distribution is given by Eq. (1-7), where A is the area of the aperture. Chapter 4 develops amplitude taper efﬁciency for nonuniform aperture amplitude distributions to calculate the gain reduction. Phase error efﬁciency gives the gain reduction due to phase anomalies. Each of these is found from the distribution of aperture ﬁelds. Figure 2-2 plots the pattern of a half cosine aperture distribution as a dashed curve. The distribution peaks in the center and tapers linearly to zero at the edges. Tapering the aperture distribution widens the beamwidth and reduces both gain and sidelobe levels. The pattern beamwidth is 1.342 times wider than the uniform distribution beamwidth. A cosine distribution produces a −0.91-dB amplitude taper loss, and the distribution edge taper causes the sidelobes to fall off at a faster rate. Example Compute the length of the aperture with a uniform distribution that will give a 10◦ beamwidth. 50.76◦ a = 5 wavelengths λ 10◦ We can calculate radiated power by integrating the Poynting vector magnitude over the radiation sphere, but there is an easier way. We assumed that the aperture ﬁelds are free-space waves. The total power radiated is in the aperture, Pr |E|2 Pr = ds Pavg = Uavg = η 4π aperture

where η is the impedance of free space. The radiated electric ﬁeld is e−j kr (1 + cos θ ) f(kx , ky ) 2λr The Poynting vector magnitude is E=j

Sr =

(1 + cos θ )2 |E|2 = |f(kx , ky )|2 η 4λ2 r 2

(2-15)

By combining Eqs. (2-14) and (2-15), we determine directivity: Sr r 2 U (θ, φ) = Uavg Pr /4π 2 j k·r Ee ds π(1 + cos θ ) = λ2 |E|2 ds

directivity(θ, φ) =

(2-16)

By considering electric and magnetic ﬁelds separately in the aperture, we eliminate the requirement that the ratio electric and magnetic ﬁelds are the same as free space used in the Huygens source approximation. Given the ﬁelds in an aperture, we can equate

APERTURES: HUYGENS SOURCE APPROXIMATION

55

them to magnetic and electric currents: Ms = E a × n

Js = n × Ha

(2-17)

where Ea and Ha are the aperture ﬁelds and n is the outward normal. The equivalence theorem [2, p. 113] results in exact solutions by using the total aperture ﬁeld, incident and reﬂected. When using the equivalence theorem, we replace the total ﬁelds present with equivalent currents. The induction theorem equates currents only to the incident ﬁelds on the aperture, which ignores wave reﬂection and results in approximate solutions: Ms e−j k|r−r | Js e−j k|r−r | ds ds A=µ (2-18) F=ε 4π|r − r | 4π|r − r | s

s

We derive the radiated ﬁelds from each distribution of currents by using vector potentials where r is the ﬁeld point and r is the source point in the aperture. These expressions are valid in the near and far ﬁelds. By integrating over only a ﬁnite aperture, we assume zero ﬁelds outside the aperture, while rigorous expressions require integrals over closed boundaries. A planar aperture must extend to inﬁnity, but the ﬁelds outside the aperture are nearly zero and contribute little. 2-2.1 Near- and Far-Field Regions The radiative near- and far-ﬁeld regions are characterized by the approximations made to the integrals [Eq. (2-18)]. The radiative near-ﬁeld region lies between the near ﬁeld, with no approximations, and the far-ﬁeld region. In both approximations the ﬁeld (observation) distance r is substituted for |r − r | in the amplitude term. The vector potentials reduce to µ ε −j k|r−r | Ms e ds A= Js e−j k|r−r | ds (2-19) F= 4πr 4πr s

s

We handle the phase term differently in the two regions. First, we expand the phase term in a Taylor series, 1 |r − r | = r 2 + r 2 − 2r · r = r − rˆ · r + [r 2 − (ˆr · r )2 ] · · · 2r where rˆ is the unit vector in the ﬁeld point direction. We retain the ﬁrst two terms for the far-ﬁeld approximation and the vector potentials become e−j kr ε F= Ms ej k·r ds , etc. (2-20) 4πr s

where we have combined k, the propagation constant, with the unit vector rˆ : k = kˆr = k(sin θ cos φ xˆ + sin θ sin φ yˆ + cos θ zˆ ) The magnetic vector potential integral parallels Eq. (2-20) as in Eq. (2-19). In the radiative near-ﬁeld zone approximation the terms in r 2 are retained and we obtain the following integral for the electric vector potential: kr 2 (k · r )2 e−j kr ε Ms exp j (k · r ) + − (2-21) F= ds 4πr 2rk 2r

56

RADIATION STRUCTURES AND NUMERICAL METHODS

No clear boundary between the three regions exists because the ﬁelds are continuous. Common boundaries are r <1 near ﬁeld L r L 1< < radiative near ﬁeld L λ L r > far ﬁeld L λ where L is the maximum dimension on the aperture. Example Determine the maximum difference between the radiative near- and far-ﬁeld approximations at a point normal to the maximum aperture dimension when r = L2 /λ and r = 2L2 /λ. Normal to the maximum dimension, rˆ · r = 0. The phase difference is

2 kr max 2r

= where rmax

phase difference φ =

L 2

2πL2 8λr

φ = π/4 at r = L2 /λ

and φ = π/8 at r = 2L2 /λ

The usual minimum distance used for antenna patterns is 2L2 /λ, where L is the maximum dimension of the antenna. At that distance, the phase error across the aperture from a point source antenna is π/8. The distance is not sufﬁcient for low-sidelobe antennas [3] because quadratic phase error raises the measured sidelobes. We can use vector potentials in the aperture after determining equivalent currents, but we will ﬁnd it more convenient to use the ﬁelds directly. Deﬁne the following integrals: j k·r Ea e ds g= Ma ej k·r ds (2-22) f= s

s

using the far-ﬁeld approximation. Near-ﬁeld integrals require additional phase terms. Given an aperture, we calculate the vector potentials in terms of Ea and Ha through the currents by using either the equivalence or inductance theorems, and we use the integrals of Eq. (2-22) in the vector potentials. We combine the ﬁelds in the far ﬁeld due to each partial source: E = −j ωA − j ηωF × rˆ For an aperture in the x –y plane, we carry out these steps by using the inductance theorem and obtain the following far ﬁelds from the incident aperture ﬁelds j ke−j kr [fx cos φ + fy sin φ + η cos θ (−gx sin φ + gy cos φ)] 4πr −j ke−j kr Eφ = [(fx sin φ − fy cos φ) cos θ + η(gx cos φ + gy sin φ)] 4πr Eθ =

(2-23)

BOUNDARY CONDITIONS

57

where f and g have been expanded in terms of their x- and y-components and η is the impedance of free space. 2-2.2 Huygens Source The Huygens source approximation is based on the assumption that the magnetic and electric ﬁelds are related as in a plane wave in the aperture: ηgy = fx

and

− ηgx = fy

ηHy = Ex

and

− ηHx = Ey

since

With this approximation, the far ﬁeld [Eq. (2-23)] becomes j ke−j kr (1 + cos θ )(fx cos φ + fy sin φ) 4πr −j ke−j kr Eφ = (1 + cos θ )(fx sin φ − fy cos φ) 4πr Eθ =

(2-24)

The two-dimensional vector Fourier transform f = (fx , fy ) of the aperture electric ﬁeld in the x –y plane determines the far-ﬁeld components. We derive the radiated components by projecting (vector scalar product) this ﬁeld onto the vectors θˆ / cos θ ˆ The transform f expands the ﬁeld in k-space [usually, (kx , ky )]. This normalizes and φ. the pattern and removes the direct dependence on aperture length. We separate out all but f when we consider aperture distributions. We drop the terms for the radiation from a point source and the pattern of a Huygens point source [Eq. (2-14)] and limit our discussions to Huygens sources and far ﬁelds. General aperture ﬁelds require Eq. (2-23), and for any region other than the far ﬁeld, additional phase terms are needed in the transforms [Eq. 2-21)]. 2-3 BOUNDARY CONDITIONS Material boundaries cause discontinuities in the electric and magnetic ﬁelds. The effects can be found by considering either vanishing small pillboxes or loops that span the boundary between the two regions. By using the integral form of Maxwell’s equations on these differential structures, the integrals reduce to simple algebraic expressions. These arguments can be found in most electromagnetic texts and we give only the results. Conversely, we will discover that artiﬁcial boundaries such as shadow and reﬂection boundaries used in geometric optics (ray optics) cannot cause a discontinuity in the ﬁelds because they are not material boundaries. The idea that the ﬁelds remain continuous across the boundary leads to the necessity of adding terms to extend ray optics methods. We discuss these ideas when considering the uniform theory of diffraction (UTD) method used with ray optics. Suppose that we have a locally plane boundary in space described by a point and a unit normal vector nˆ that points from region 1 to region 2. We compute the tangential ﬁelds from the vector (cross) product of the ﬁelds and the normal vector. The ﬁelds

58

RADIATION STRUCTURES AND NUMERICAL METHODS

can be discontinuous at the interface between the two regions if surface magnetic MS or electric current JS densities exist on the surface. nˆ × (E2 − E1 ) = −MS

nˆ × (H2 − H1 ) = JS

(2-25a,b)

The normal components of the ﬁelds change due to the differing dielectric and magnetic properties of the materials and the charges induced on the surface: nˆ · (ε2 E2 − ε1 E1 ) = ρS

nˆ · (µ2 H2 − µ1 H1 ) = τS

(2-26)

with ρS and τS given as electric and magnetic surface charge densities, respectively. Perfect dielectric and magnetic materials can have no currents, which reduces Eq. (2-25) to nˆ × (E2 − E1 ) = 0 nˆ × (H2 − H1 ) = 0 (2-27) Equation (2-27) means that the tangential ﬁelds are continuous across the boundary. These boundary conditions are used in the method of moment analyses to determine currents. The method applies the boundary condition in integral equations to determine the coefﬁcients of the expansion of currents in the sum of basis functions. The currents described as these sums do not satisfy the boundary conditions at all points but do when integrated over a region. This method leads to approximations that will converge as more terms are included in the expansions. When doing analysis we ﬁnd two types of surfaces convenient. We use these surfaces to reduce analysis effort by using planes of symmetry. The ﬁrst one is the perfect electric conductor (PEC). A PEC surface causes the ﬁelds to vanish inside and to have electric currents induced on it: nˆ × E2 = 0

nˆ × H2 = JS

(PEC)

(2-28a,b)

A PEC surface is also called an electric wall. The second surface is the perfect magnetic conductor (PMC) and is a hypothetical surface. Whereas good conductors approximate PEC, there are no PMC materials. The PMC has no internal ﬁelds like the PEC and forces the tangential magnetic ﬁeld to be zero: nˆ × E2 = −MS

nˆ × H2 = 0

(PMC)

(2-29)

A PMC surface supports the hypothetical magnetic current density MS . We ﬁnd that the magnetic wall (PMC) concept simpliﬁes analysis.

I

I

I

M V

I

I

I Electric Currents

FIGURE 2-3

M

M

+ − + −

V Voltage Source

M

M

M

Magnetic Currents

Ground-plane images.

PHYSICAL OPTICS

59

We use images of currents to include material boundaries in analysis. Figure 2-3 illustrates ground-plane images. When we analyze radiation from currents in the presence of a boundary, we include the actual antenna and its image to compute the ﬁelds. The ﬁgure shows an inﬁnite ground plane, but a ﬁnite ground-plane image can be used in the angular region where a reﬂected wave occurs in the ﬁnite plane. We consider this idea further when discussing geometric optics. We can use images in dielectric boundaries provided that we calculate the polarization sensitive reﬂection coefﬁcients to adjust the magnitude and phase of the image. 2-4 PHYSICAL OPTICS Physical optics uses things that can be measured. We can measure both currents and ﬁelds, but auxiliary vector potentials have no physical reality, only mathematical artifacts that simplify Maxwell’s equations. Nevertheless, the auxiliary vector potentials provide simple models for problems that enable simple mental pictures, as shown earlier, but we cannot easily formulate them into a systematic analysis tool for antenna problems. The physical optics analysis method combines the use of Green’s functions to calculate ﬁelds radiated by a given distribution of currents and then uses boundary conditions to determine the currents induced on objects due to incident ﬁelds. We compute the effects of a mounting structure by inducing currents on it and adding their radiation to the antenna pattern. The method assumes that radiation from the induced currents on the structure does not change the initial currents. We start analyses from either currents or incident ﬁelds and work from those. The resonant structure of many antennas determines the approximate current distribution that we normalize to the radiated power. We calculate the ﬁelds from these currents. Physical optics can use an iterative technique to calculate incremental currents induced on the original radiators and improve the solution, but we usually just sum the radiation from the original currents to the radiation from the induced currents. The second starting point for physical optics can be incident ﬁelds. These could be plane waves or could be ﬁelds found from the measured radiation patterns of antennas: for example, the pattern of a reﬂector feed. We add the radiation from the induced currents to the incident waves. 2-4.1 Radiated Fields Given Currents The radiated ﬁelds can be found from distribution of the electric and magnetic currents by the use of dyadic Green’s functions that contain source and ﬁeld coordinates. We sometimes refer to the Green’s functions as vector propagators or transfer functions between currents and ﬁelds. We calculate the ﬁelds from integrals over the source points of the dot (scalar) product between the dyadic and current densities. The dyadic Green’s function contains both near- and far-ﬁeld terms and requires slightly different expressions for the electric and magnetic ﬁelds. The general propagator from electric and magnetic currents has separate terms for electric and magnetic currents, which when used with surface patch currents can be reduced to short subroutines or procedures easily programmed [1]: (2-30) E(r) = GEJ (r, r ) · J(r ) dV + GEM (r, r ) · M(r ) dV H(r) = GHJ (r, r ) · J(r ) dV + GHM (r, r ) · M(r ) dV (2-31)

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RADIATION STRUCTURES AND NUMERICAL METHODS

These expressions integrate over the currents located at source points r for a dyadic Green’s function that changes at each ﬁeld point r and source point r . Although these Green’s functions are valid at all ﬁeld points in space both near and far ﬁeld, they are singular at a source point. Only retaining terms with 1/R dependence for the far ﬁeld greatly simpliﬁes the expressions. When ﬁelds are incident on a perfect electric conductor (PEC), the combination of incident and reﬂected tangential magnetic ﬁelds induces an electric current density on the surface. The ﬁelds inside the conductor are zero. We assume locally plane surfaces on patches and compute currents that satisfy the boundary condition. Given the local unit normal nˆ to the surface, the induced current density is given by JS = nˆ × (Hincident + Hreﬂected ) Hincident = Hreﬂected JS = 2nˆ × Hincident

(2-32)

The reﬂected magnetic ﬁeld equals the incident magnetic ﬁeld because the ﬁeld reﬂects from the conductive surface. The sum of the tangential electric ﬁelds must be zero. Because the reﬂected wave changes direction, the vector (cross) product of the electric and magnetic ﬁelds must change direction. The reﬂected tangential electric ﬁeld changes direction by 180◦ , so the tangential magnetic ﬁeld must not change direction because the Poynting vector changed its direction. Equation (2-32) is the magnetic ﬁeld equation applied on a PEC. Equation (2-25b) is the general magnetic ﬁeld equation at a boundary. Physical optics starts with a given current distribution that radiates, or the measured pattern of an antenna. When an object is placed in the radiated ﬁeld, the method calculates induced current on the object to satisfy the internal ﬁeld condition. For example, PEC or PMC have zero ﬁelds inside. When we use simple functions such as constant-current surface patches, the sum of the radiation from the incident wave and the scattered ﬁelds from induced surface currents produces only approximately zero ﬁelds inside. As the patch size decreases, the method converges to the correct solution. To obtain the radiated ﬁeld everywhere, we sum the incident wave and scattered waves. The ﬁelds radiated by the induced currents produce the shadow caused by the object. With geometric optics techniques such as UTD, the object blocks the incident wave and we determine the ﬁelds in the shadow regions from separate diffraction waves. In physical optics the incident wave continues as though the object were not present. Only geometric optics techniques use blockage. We can calculate the ﬁelds radiated from antennas in free space or measure them in an anechoic chamber that simulates free space, but we mount the antenna on ﬁnite ground planes, handsets, vehicles, over soil, and so on, when we use them. Physical optics is one method of accounting for the scattering. We show in later chapters that the mounting conﬁguration can enhance the patterns. 2-4.2 Applying Physical Optics In this book we do not discuss how to develop numerical techniques, but it is important to understand how to apply methods. Whether you develop your own codes or use commercial codes, certain rules should be applied. Consider Eq. (2-32). The normal to

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61

the surface points in the direction of the incident wave: outward. If the normal pointed inward, the sign of induced electric current density would change. Most codes have made the assumption that the normal points outward, but some codes may check on the direction of the normal relative to the incident wave and make the necessary sign change. We must keep track of the direction of the normal, and it may be necessary to rotate the normal depending on the expected direction of the incident wave. If an object can have radiation from both sides, it may be necessary to use two objects in the analysis. Many codes store each object as a separate entity in a disk ﬁle. In some cases we need to store an object multiple times. Take, for example, a Cassegrain dual reﬂector. The feed antenna illuminates the subreﬂector and induces currents on it. These currents radiate and excite currents on the main reﬂector. When the main reﬂector–induced currents radiate, the subreﬂector intercepts or blocks part of the ﬁelds. We account for this blockage by using a second subreﬂector object on which the code calculates a new set of induced currents by using the main reﬂector currents as the source. We could add these currents to the existing disk ﬁle object or merely keep the second object. We want to keep the second object separate so that we can calculate additional currents induced on the main reﬂector using these currents as sources. These currents will be reduced from the initial set, but they are an important contribution to the ﬁelds radiated behind the reﬂector. This example illustrates iterative PO. When objects face each other signiﬁcantly, iterative PO is necessary to calculate correct patterns. The method converges rapidly in most cases. Figure 2-4 illustrates the geometry of a corner reﬂector. A half-wavelength-long dipole is placed between two metal plates usually bent to form a 90◦ angle. We can use other angular orientations between the plates, but this is the usual design. The ﬁgure does not show the feed line to the dipole, which usually starts at the juncture of the two plates and runs up to the dipole. This feed line contains the balun discussed in Section 5-15. Although the ﬁgure shows the plates as solid, many implementations use metal rods to reduce weight and wind loading. The analysis starts with assumed currents on the dipole. We divide the plates analytically into small rectangular patches, which can be small (≈ λ/8 to λ/4) on a side since it takes only a few to cover the plates. You should repeat the analysis with

FIGURE 2-4

Corner reﬂector with a dipole located between two ﬂat plates.

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RADIATION STRUCTURES AND NUMERICAL METHODS

different-sized current patches to determine if the analysis has converged. In a similar manner, we break down the current on the dipole into short linear segments, each with constant amplitude. By using a near-ﬁeld version of Eq. (2-31), we calculate the magnetic ﬁeld incident on each patch on the plates. This ﬁeld induces electric currents on the plates calculated from Eq. (2-32). Remember that we combine radiation from the source dipole with that radiated from the induced currents to reduce the radiation behind the antenna. The currents were induced to satisfy the boundary condition of the plate, but only with both radiations present. Figure 2-5a illustrates this process of inducing currents. Figure 2-6 shows the antenna pattern calculated using these currents. The E-plane pattern drawn as a solid line produces a null at 90◦ because the dipole pattern has this null. The plates cause the narrowing of the beam in the H -plane. The plates reduced the back radiation to −22 dB relative to the forward radiation, called the front-to-back ratio (F/B). The gain has increased from the 2.1 dB expected from a dipole to 9.3 dB. An equivalent geometric optics analysis uses two images in the plates, as shown in Figure 2-5b, for the analysis. If you look at Figure 2-4 or 2-5, you should notice that the two plates face each other. Currents on one plate will radiate toward the other plate and induce another set of currents on it. We could ignore these induced currents if the radiation was insigniﬁcant, but to produce correct patterns we must include them. The solution to this problem calls for an iterative technique where we calculate the radiation from the currents on the ﬁrst plate and induce incremental currents on the second plate. These incremental currents produce further radiation that induces additional currents on the other plate. The method converges rapidly. Figure 2-7 gives the antenna pattern after the iterations have been completed and we include radiation from all currents. The actual F/B ratio of the antenna is 29 dB, and the additional currents increased the gain by 0.7 dB to 10 dB. Adding the two plates in the original analysis increased the gain by 7.2 dB, whereas the iterative technique had a much smaller effect. Figure 2-8 illustrates the iterative technique and shows that the equivalent geometric optics analysis adds a third image to represent the reﬂection between the plates. Remember when you mount the antenna in an application, the structure will change the realized pattern, but the high F/B ratio reduces this effect. The mounting structure used when measuring the antenna changes the pattern as well, which limits our knowledge of the real pattern.

Dipole Currents

Image (a)

Image (b )

FIGURE 2-5 Cross-sectional view of a corner reﬂector: (a) magnetic ﬁeld radiated from a dipole induces currents on plates; (b) plate currents replaced with image dipoles.

PHYSICAL OPTICS

E-plane

63

H-Plane

FIGURE 2-6 Pattern calculated from a combination of dipole and plate currents in a corner reﬂector with 1 × 0.9λ plates without induced current iteration.

Physical optics can determine the impedance effects of the limited images in the ground planes, such as the corner reﬂector. The local nature of impedance effects allows the use of images to calculate the mutual impedance effects of ground planes. We use impedance calculations not only to determine the bounds of ground-plane effects on input impedance, but to calculate the total power radiated by the antenna. The images (excited currents on ground planes) radiate but do not receive input power. A ground plane at least λ/2 on a side located about λ/4 away from the antenna produces nearly the same impedance effects as an inﬁnite ground plane, but the ground plane alters the radiation pattern greatly because it restricts possible radiation directions. It has commonly been thought that physical optics could compute the ﬁeld only in the main beam pattern direction of a paraboloidal reﬂector. The method can determine this pattern region accurately by using only a few patches, each one being many wavelengths on a side. As the processing power of computers increases, the patch size can be shrunk until PO can calculate the pattern in every direction, including behind the reﬂector. It is important to remember to include the feed pattern behind the reﬂector even though its radiation is obviously blocked by the main reﬂector. Physical optics uses induced currents to cancel the ﬁelds inside objects when the incident ﬁelds and the radiation from the induced currents are added. We can calculate the pattern behind a reﬂector using UTD (GTD), the uniform (geometric) theory of diffraction. This

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E-Plane

H-plane

FIGURE 2-7 Pattern of corner reﬂector with 1 × 0.9λ plates with induced current iteration equivalent to multiple reﬂectors between the plates.

Dipole

Image

Image

( a)

Image (b)

FIGURE 2-8 (a) Wall currents on plates radiate magnetic ﬁelds that induce additional currents on facing plates; (b) added induced currents equivalent to additional image dipole.

geometric optics-based method blocks the radiation from the feed and uses diffractions from the rim edge to calculate the pattern behind the reﬂector. We discuss UTD in Section 2-7. A comparison of UTD and physical optics calculations [1,4] of the pattern behind shows that the two methods match. The dashed curve of Figure 2-9 plots the results of the PO analysis of a 20λ-diameter centrally fed paraboloidal reﬂector. The feed antenna radiation tapers to −12 dB at the

65

Gain, dB

PHYSICAL OPTICS

Pattern Angle

FIGURE 2-9 Physical optics analysis of a 20λ-aperture-diameter paraboloidal reﬂector (dashed curve) compared to analysis that includes PTD (solid curve).

reﬂector rim. Figure 2-9 shows the feed power spillover peaking at angles off the boresight near 100◦ . PO analysis computes the currents on a patch by assuming that it is embedded in an inﬁnite plate. The reﬂector rim violates this assumption and we need extra terms to calculate the pattern behind the reﬂector accurately. Adding PTD (the physical theory of diffraction) to PO improves the match between the two methods behind the reﬂector as shown by the solid curve on Figure 2-9. PTD handles caustic regions of PO in a manner similar to the equivalent current method based on diffraction coefﬁcients of UTD with geometric optics for shadow and reﬂection boundaries. For this example, the additional PTD currents add with the same phase because of the symmetry of the reﬂector geometry and produce the maximum effect. The PTD currents on the rim of an offset reﬂector will not add and produce a peak effect behind the reﬂector but will produce a more defuse effect. We only need PTD over a limited pattern angular range to reduce error, and the cost of implementing the ﬁx may exceed the necessity of knowing the pattern in these regions. Similarly, UTD needs the addition of edge currents for accurate calculation of the radiation near 180◦ , behind the reﬂector. Although any model for the feed pattern can be used with PO, results matching UTD exactly occur only over all regions of the back radiation when the feed satisﬁes Maxwell’s equations in the near and far ﬁelds [4, p. 212]. One such feed is the Gaussian beam approximation. Again, like PTD ﬁxes, the small errors when using other feed antenna approximations occur only at limited pattern regions that may be unimportant. 2-4.3 Equivalent Currents We can relate the concept of equivalent currents to physical optics. In this case we generate an artiﬁcial surface that covers a source of radiation. The incident ﬁelds generate surface electric and magnetic current whose radiation cancels the internal

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ﬁelds and generates the external pattern. We use these at the apertures of antennas such as horns. By using the dyadic Green’s functions we can calculate the near-ﬁeld patterns and the coupling between antennas when the assumption is made that the presence of a second antenna does not alter the aperture ﬁelds. Given the outward normal n, ˆ we calculate the equivalent currents by nˆ × Eincident = −MS

nˆ × Hincident = JS

(2-33)

We must use both electric and magnetic current densities on the surface to replace the internal ﬁelds. If the ratio of the electric ﬁeld to the magnetic ﬁeld equals the impedance of free space (376.7 ), the combination of the two currents produces the radiation of the Huygens aperture source when used with the dyadic Green’s function. We use equivalent currents for a variety of analyses over ﬂat apertures such as horns and paraboloidal reﬂectors, but they can also be used with curved structures or apertures. We can, for example, use equivalent currents for calculation of the effects of radomes. Locally, we assume that the incident waves are plane waves and use boundary conditions to calculate reﬂected and transmitted waves. It is necessary to separate the incident wave into parallel and perpendicular polarizations, the ray-ﬁx representation discussed in Section 2-7.8. These polarizations have differing reﬂection and transmission coefﬁcients. We generate one surface on the inside of the radome and another on the outside. We use locally free-space waves for the reﬂected and transmitted waves lying outside the radome. Both these waves can be replaced with equivalent currents. The equivalent currents produce null ﬁelds inside the radome when combined with the incident wave radiation [4, p. 155]. Including these equivalent currents in a PO analysis, we add the effect of the radome. Equivalent currents can also be used with lenses. We use the incident waves combined with the idea of locally plane waves to calculate reﬂected and transmitted waves at each surface and replace them with equivalent currents. We include the dielectric constant of the lens in the dyadic Green’s functions for the internal radiation of the lens to calculate the ﬁelds at the second surface. We apply locally plane waves at the second surface to determine the transmitted and reﬂected rays and then replace them with equivalent currents. Because the lens has internal reﬂections, we need to apply an iterative PO analysis to calculate the multiple reﬂections between the two surfaces. The method converges rapidly because the internal reﬂections are small. 2-4.4 Reactance Theorem and Mutual Coupling In Section 1-14 we discussed how the coupling between two antennas can be found from reactance. Given a transmitting antenna that generates a ﬁeld at the receiving antenna, the reactance is described by an integral equation [5]: reactance =

(Et · Jr − Ht · Mr ) dV = t, r

(2-34)

The volume integral is over the receiving antenna currents, but it is often reduced to a surface or line integral. A second form of Eq. (2-34) uses the ﬁelds radiated by

METHOD OF MOMENTS

67

both antennas. Given a surface that surrounds the receiving antenna, the integral for reactance is taken over this surface: (Er × Ht − Et × Hr ) · ds = t, r

reactance =

(2-35)

Sr

The differential normal ds is pointed away from the receiving antenna. When we represent the two antennas and the transmission between them as an impedance matrix, it implies that we know the input currents to both antennas. By expressing the coupling as an impedance matrix, we compute mutual impedance from the reactance integral: −1 t, r (2-36) Z12 = I1 I2 Antennas that we describe by input currents only have electric current densities excited on their surfaces. The mutual impedance formula using reactance reduces to Z12 =

−1 I1 I2

Et · Jr dV

(2-37)

Vr

The volume integral reduces to a line integral in most cases. Antennas with given input voltages such as slots can be described using magnetic currents and we use a mutual admittance matrix for the antenna pair: Y12 =

1 −1 · reactance = Vt Vr Vt Vr

Ht · Mr dV

(2-38)

Vr

By using reciprocity antennas made of linear, isotropic materials, we have equal crossmatrix terms: Z12 = Z21 and Y12 = Y21 (2-39) We calculate self-impedance terms by integrating over the surface of the antenna: for example, the radius of a dipole with the source of the ﬁeld located at the center of wires or slots. 2-5 METHOD OF MOMENTS The method of moments (MOM) [6] expands the currents on an antenna (or scattering object) in a linear sum of simple basis functions. The approximate solution is a ﬁnite series of these basis functions: N fa = ai fi (2-40) i=1

We compute the coefﬁcients by solving integral equations to satisfy boundary conditions on the surface of the antenna (or object). The integral equation can be expressed in the form Lfa = g, where L is a linear operator, usually a scalar product using an integral, fa the unknown currents given by Eq. (2-40), and g the known excitation

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or source function. We substitute the summation of Eq. (2-40) into the linear operator equation and use the scalar product integral to calculate the terms in a matrix equation. The solution of the matrix equation determines the coefﬁcients of current expansion. The MOM produces ﬁlled matrices that require time-consuming numerical methods for inversion. The art of the MOM is in choosing basis functions and deriving efﬁcient expressions for evaluating the ﬁelds using the basis function currents. Common basis functions are simple staircase pulses, overlapping triangles, trigonometric functions, or polynomials. The method does not satisfy boundary conditions at every point, only over an integral average on the boundaries. By increasing the number of basis functions, the method will converge to the correct solution. We need to judge how many terms are required for an adequate engineering evaluation. Spending excessive time on the solution cannot be justiﬁed if it greatly exceeds our ability to measure antenna performance accurately using real hardware. 2-5.1 Use of the Reactance Theorem for the Method of Moments We can use the reactance theorem to generate a moment method solution to the currents on a thin-wire antenna. Thin-wire solutions assume that there are no circumferential currents and reduces the problem to ﬁlamentary currents. An electric ﬁeld integral equation (EFIE) satisﬁes the boundary condition of Eq. (2-25a), a zero tangential ﬁeld at the surface of the wires, but it does not seem explicit in the derivation. The reactance theorem produces an impedance matrix whose inversion yields the coefﬁcients of the current expansion [7]. Similar to many other methods, the Green’s function has been solved explicitly to reduce run time. This method [7] uses overlapping sinusoidal currents on V-dipoles as basis function currents and uses the Green’s function to calculate the radiation from one V-dipole at the location of a second V-dipole. Both the radiating and receiving dipoles use the same expansion function. Galerkin’s method uses the same weighting (or testing) function as the basis function and yields the most stable solutions. The reactance equation (2-37) calculates the mutual impedance between the two dipoles when each has unity current. We compute self-impedance by spacing a second V-dipole one radius away and by using the reactance theorem to calculate mutual impedance, a technique equivalent to the induced EMF method. The scalar (dot) product between the incident vector electric ﬁeld and the current density along the dipole reduces the vectors to scalars that can be integrated. The current density acts as the testing or weighting function for the method of moments. Performing the integration means that the current density only satisﬁes the zero tangential electric ﬁeld boundary condition in an average sense. If series impedances are placed in the V-dipole, their impedance is added to the diagonal elements of the mutual impedance matrix. To excite the structure, we place a delta voltage source in series with the Vdipole terminals. The solution for the currents can be found by inverting the matrix equation and using the voltage excitation vector starting with the matrix equation [Zmn ][Im ] = [Vn ]

(2-41)

After computing the matrix inverse and specifying the input voltage vector, the complex current values are found on the structure: [In ] = [Zmn ]−1 [Vm ]

(2-42)

METHOD OF MOMENTS

69

Given the input voltage and the solution for the currents, the input impedance can be calculated. Similarly, the far- and near-ﬁeld patterns can be calculated by using Eqs. (2-30) and (2-31) of the dyadic Green’s function. The code must satisfy Kirchhoff’s current law at the junction between groups of Vdipoles, which adds a constraint to the currents. Because an overlapping sinusoidal basis function closely follows the actual currents normally excited on dipoles, the segments can be on the order of a quarter-wavelength long or more and yield acceptable results. Basis functions that closely follow expected current distributions are sometimes called entire domain functions. These reduce the size of the matrix to be inverted but require more complicated calculations for matrix terms and radiation. Although the concept of a V-dipole was expanded to a V rectangular plate [8], the method is only a subset of general integral equation solutions. This approach generates a simple impedance matrix formulation easily understood from an engineering point of view. 2-5.2 General Moments Method Approach The method of moments can solve other types of electromagnetic problems: for example, electrostatic problems involving charges and dielectrics [9]. These solutions can determine the characteristic impedance of transmission lines useful in the design of antenna feeders. All moment method solutions are found from the solution of integral equations over boundary conditions. The boundary conditions can be either the tangential electric ﬁeld (EFIE) or magnetic ﬁeld (MFIE) conditions given by Eq. (2-25a,b) or a combination applied using an integral scalar product. We need a combination for closed bodies near an internal resonance frequency (resonant cavity) because the solutions exhibit resonances that make the solution invalid over a narrow frequency range. The method of moments can be applied to dielectric bodies when we use the constitutive relations of Eqs. (2-25) and (2-26), where the formulations for dielectric bodies use either volume or surface integrals [9]. Consider the use of the electric ﬁeld integral equation (EFIE) with metal surfaces. We expand the currents on the objects using basis functions Bm (r ) with coefﬁcients Im : J(r ) =

Im Bm (r )

(2-43)

The basis functions can be applied over a limited range of the structure in piecewise linear functions, which can be staircase pulses, overlapping triangular functions, or sinusoidal basis functions, whereas multiple functions can be applied over the whole or part of the structure for entire domain basis functions. For example, these could be a sum of sinusoidal functions which form a Fourier series representation. On a PEC surface the tangential electric ﬁeld vanishes [Eq. (2-28a)]. At ﬁeld point r along the surface S, nˆ × [Eincident (r) + Escattered (r)] = 0 M Im Bm (r ) · G(r, r ) ds Escattered = m=1

(2-44)

s

We can only satisfy Eq. (2-44) using a ﬁnite sum in the average sense of an integral. Since the integral and summation operate on a linear function, we can interchange them.

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RADIATION STRUCTURES AND NUMERICAL METHODS

We introduce weighting (or testing) vector functions tangent to the surface Wn (r) and take the scalar (dot) product of this vector with the sum of electric ﬁelds. This limits the result to the tangential component of the electric ﬁeld: [Eincident (r) · Wn (r) + Escattered (r) · Wn (r)] ds = 0

(2-45)

s

We identify the weighted integral of the incident ﬁeld with the source and weighted integral of the ﬁeld radiated by the basis functions (scattered ﬁeld) as the impedance matrix terms. The integrals over the boundaries are one form of scalar product represented by · notation. Using unity current on each basis function, we calculate the matrix terms by using the scalar product:

Bm · G(r, r ) ds , Wn (r) =

Zmn = s

Vn = −Eincident (r), Wn (r) = −

Bm · G(r, r ) · Wn (r) ds ds (2-46)

s

s

Eincident (r) · Wn (r) ds

(2-47)

s

The combination of Eqs. (2-46) and (2-47) when integrated over each portion of the source gives a matrix equation: [Zmn ][Im ] = [Vn ]

(2-48)

The weighting functions could be as simple as pulse functions, overlapping triangular functions on lines or surfaces (rooftop), piecewise sinusoidal functions, or others. The type of basis functions determines the convergence more than the weighting (testing) functions, which only determine the averaging. Realize that the moment method converges to the exact solution when we increase the number of basis functions, but it is a matter of engineering judgment to determine how many terms give acceptable answers. Equation (2-47) deﬁnes the source voltage occurring over a segment when the formulation uses a piecewise function expansion. The incident voltage is the weighted integral of the incident electric ﬁeld. For example, the NEC formulation applies an excitation voltage across one segment. The reaction integral formulation of Section 25.1 applies a voltage source at the end of a segment. The modeling of sources is an important part of the art in the method of moments. The expansion of Eq. (2-44) is only one possible moment method solution. We could use the boundary condition on the magnetic ﬁeld, a combination of the electric and magnetic ﬁeld conditions on a PEC. If the surface has ﬁnite conductivity, the boundary conditions are modiﬁed. The moment method is a general method that computes approximate solutions to the currents. Unlike physical optics, the currents do not have to be assumed beforehand but are found as a ﬁnite series approximation. Antenna designers discover that adequate codes are available for most problems. Moment method solutions are typically limited to objects only one or two wavelengths in size, although any method can be stretched. Analysis of large structures becomes intractable because of the large amount of computer memory required and the length

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71

of time needed to calculate the solution. Coarse models may not give totally accurate results but can be useful in determining trends. Given these ideas, remember that physical models can be built that solve the electromagnetic problem instantaneously. We found that it takes considerable time to learn any code, and a new code has to offer considerable advantages or solve problems that the present one cannot solve before we invest our time. 2-5.3 Thin-Wire Moment Method Codes Thin-wire codes that assume only ﬁlamentary currents are readily available. We have experience with NEC, the Richmond code (ASAP), and AWAS [10], a commercial code. All have advantages, but they take time to learn. A commercial code with a graphical interface makes the input and output easier: for example, for NEC. These pay for themselves quickly by saving time. NEC can include plates, but since it uses a MFIE (magnetic ﬁeld integral equation) for them, it is limited to closed bodies. When accuracy becomes important, it is necessary to decrease the segment length and increase their number. These codes use matrix inversion with calculation time proportional to N 3 and a matrix ﬁll time proportional to N 2 . Run time increases enormously as the number of segments increases. The commercial code AWAS determines the segmentation, while the user of NEC must specify it. The rule is to use at least 10 segments per wavelength, but initial analysis can tolerate the errors due to using fewer segments. The segments should be longer than the diameter, and care must be taken that the segments do not overlap because the radius of the wires is too large. Solid objects, such as plates, can be modeled as wire frames, with the rule that the perimeter of the wire equal the spacing between the wires [11]. This rule can be violated, but a test of the convergence should be made. When we model slots in a solid object, we cannot apply the perimeter equalspacing rule because the slot will disappear. These codes compute the radiation pattern more accurately than the input impedance due to simplistic source models, and we may have to build the antenna to determine the true input impedance. Of course, an antenna with a good input impedance response that does not have the required pattern is useless. We can reduce NEC run time if the antenna has symmetry with multiple inputs. The code reduces input by allowing the user to specify symmetry. For example, a multiarm spiral analysis requires only the input of one arm. The various mode voltages are entered after the basic structure impedance matrix has been solved. If an object has M-way symmetry, the matrix ﬁll time is reduced by M 2 and the solution time by M 3 . The various voltage modes can be applied afterward. If we add another wire segment after specifying symmetry, the symmetry is destroyed and the program uses the full matrix. The only advantage we gain is in specifying the model because the program solves the full matrix instead of the reduced matrix. 2-5.4 Surface and Volume Moment Method Codes Antennas made of plates or containing ﬁnite plate ground planes can be solved by using wire meshing of a thin-wire code. The method of moments code has been extended to plates [12,13] using a rooftop basis function on both rectangular and triangular patches. The number of basis functions (i.e., matrix size) grows rapidly. One solution is to use entire domain basis functions. These require more complicated integrals, but

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they reduce the matrix size. Dielectric portions of the problem lead to either volumetric integrals or various forms of surface integrals that use equivalent currents to replace the internal ﬁelds [9,14]. These problems lead to a variety of boundary conditions solved using a ﬁnite series of basis function and integral equations to satisfy those boundary conditions approximately. MOM analysis of antennas mounted on dielectric substrates requires special techniques. Commercial codes determine the currents ﬂowing on these antennas while accounting for the dielectric. Often, Green’s functions are found numerically, which increases the execution time. Since the currents are located on the surface and the integrals of the boundary conditions are over the same surface, the singularity of the Green’s function causes a numerical problem. For example, the free-space Green’s function has the term 1/|r − r |, which becomes inﬁnite on the surface. Spectral domain methods remove the singularity by using a sum of current sheets on the surface as an entire domain basis function. A uniform plane wave propagating at an angle to the surface excites the current sheet. The actual current ﬂowing on the metal portions is expanded as a sum of these current sheets [15, p. 208ff; 16]. The uniform current sheets are expanded in a spatial Fourier transform as well as the Green’s function, and the MOM problem is solved. The Fourier-transformed Green’s function no longer has the singularity. When the metallization can be expressed as an inﬁnite periodic structure, the current is expanded as a Fourier series. The inﬁnite periodic structure is used with frequency-selective surfaces and inﬁnite arrays. In this case the ﬁelds and currents are expanded in Floquet modes (harmonics). 2-5.5 Examples of Moment Method Models Figure 2-10 demonstrates the use of a wire mesh to replace a solid plate. We located a resonant (≈ λ/2) dipole λ/4 distance over a λ-wide ground plane in the H -plane and offset 3/8λ from one edge. This is repeated in Figure 2-20 using GTD analysis. The rods only run parallel to the dipole because cross wires do not have currents induced on them in the ideal world of analysis. The circumference of the rods equals the spacing between the rods and forms an equivalent solid plate. An actual antenna could use smaller-diameter rods and work as effectively as the solid plate and would reduce weight and wind loading. NEC analysis produces the same pattern as the GTD analysis of Section 2-7.2, except that the E-plane size of the rods alters the backlobe predicted by GTD to some extent, because that analysis assumes inﬁnite-length rods.

FIGURE 2-10 Use of a wire mesh to replace a solid plate for dipole over a ground plane in a MOM calculation.

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73

Over most of the pattern angles the two analyses produce identical results. The NEC analysis accounts for the mutual impedance between the dipole and its image in the ﬁnite ground plane. For impedance calculations a small ground plane gives almost the same reaction to the antenna as an inﬁnite ground plane. Figure 2-11 shows a wire frame model of a cell phone. The model contains more wires than necessary for λ/10 spacing, but more wires improve the geometry match. When using crossed wires that shield both polarizations, we reduce the wire circumference in half since the wires approach the squares from four sides. The small wire antenna must be connected to the wire grid of the model to generate proper currents on the box. Either we restrict possible locations of the antenna or we must distort the wire grid locally. You should write an automatic grid generator if you use this analysis often. Consider that you need to specify whether an edge wire should be generated when two plates share the same edge. The hand holding the cell phone and the head nearby have signiﬁcant effect on the antenna performance. The model given in Figure 2-11 has limited use. We need either a moment method analysis, such as WIPL-D, which includes volume dielectric structures, or FDTD, which can include complex material structures to model the head and produce good results. Figure 2-12 illustrates a wire frame model of an airplane used for low-frequency analysis. Antennas mounted on free-ﬂying models such as airplanes or spacecraft will excite the structure. Electrically, small antennas can excite the entire vehicle as an antenna. For example, a small antenna mounted on a large ground plane that would produce vertical polarization can excite the wings or fuselage and the entire system will

FIGURE 2-11 Wire frame MOM model of a cellular telephone handset with an antenna connected to the mesh.

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RADIATION STRUCTURES AND NUMERICAL METHODS

FIGURE 2-12 Wire frame MOM model of an airplane.

radiate horizontal polarization. Models similar to Figure 2-12 can eliminate surprises. The model restricts antenna mounting locations to the wire positions and may require local distortions of the grid. Moment methods can include solid plates. Figure 2-13 shows an open waveguide horn analysis that uses a combination of plates and a single-feed wire monopole [12]. Locating the monopole or a small dipole inside the waveguide produces excitation of the waveguide mode that feeds the horn. Even though the model does not necessarily produce accurate impedance information, the model accurately calculates the pattern generated by the currents excited in the walls. We can either use an aperture method

FIGURE 2-13 MOM model of a pyramidal horn using ﬂat plates fed by a small dipole. (From [14, p. 229].)

METHOD OF MOMENTS

75

PEC (a)

PMC

(b)

FIGURE 2-14 Use of electric and magnetic walls to reduce the model size in MOM analysis of a pyramidal horn: (a) PEC wall divides the horn; (b) PMC wall divides the horn.

for the horn that replaces the aperture ﬁelds or use the currents excited in the walls to calculate the pattern. Either method works for the front lobe. The moment method calculation requires signiﬁcantly greater calculation time but produces results that better match measurements in all directions. Figure 2-14 demonstrates how to reduce calculation time by using planes of symmetry in a moment method analysis. In this case the small dipole feed is separated by two equally fed closely spaced dipoles. The right–left symmetry of the antenna allows reduction of the model by half. A vertical PMC wall divides the antenna into two parts, with only one remaining in the analysis. A horizontal PEC conductor divides the remaining model in half because halfway between the dipole feed is a virtual short circuit. Figure 2-14 contains only one-fourth the size of the original problem. Since matrix inversion requires N 3 calculations for an N × N matrix, dividing the analytical model down to one-fourth size reduces this calculation by a 64 : 1 factor. This also reduces the matrix element (ﬁll time) calculations by 16 : 1. Reducing the model by using symmetry planes enables the solution of larger problems and reduces calculation time. Analyses in later chapters use the moment method to predict antenna performance. Wire frame and plate analyses determine vehicle and mounting structure pattern effects. The moment method produces excellent analyses because it determines the approximate

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current distribution as a sum of simple basis functions and we need not start with an assumed current distribution on the antenna.

2-6 FINITE-DIFFERENCE TIME-DOMAIN METHOD The ﬁnite-difference time-domain (FDTD) method solves the coupled Maxwell’s curl equations directly in the time domain by using ﬁnite time steps over small cells in space. The method reduces the differential equations to difference equations that can be solved by sets of simple equations. The method alternates between the electric and magnetic ﬁelds solved at locations a half-step apart because central differences are used to approximate derivatives. A 1966 paper by Yee [17] described the basic method that many authors have improved upon, but the original method remains the approach of choice. FDTD can solve many types of electromagnetic problems, of which antenna analyses are only one type. Computer memory and speed limit the size of problems that can be solved, but larger and larger problems can be solved as the cost of computing keeps reducing. Besides antenna problems, the method is applied to microwave circuits, biological interaction with electromagnetic waves, optics, and radar cross-section problems. The number of uses expands every day. The method allows each cell to be made of different materials, leading to the solution of volumetric complex structures. The solution of the equations is robust and the errors are well understood. Currently, the method solves moderately small antenna problems on the order of a few wavelengths. Of course, faster and larger computers can solve larger problems, especially if the analyst has patience. FDTD handles microstrip antennas with their complex layering of dielectrics, including a ﬁnite ground plane without the use of complex Green’s functions required of frequency-domain solutions. The interaction of antennas with the near environment, such as the effect of the head on cellular telephone handsets, can be solved. In this case the complex electromagnetic properties of the head can be described as cells each with different electrical properties. In addition to giving a solution to the radiation pattern and allowing characterization of the communication system, it can provide insight into the radiation safety concerns of users. The method handles the solution of the interaction of antennas with the human body in a straightforward manner for prediction of biomedical applications, such as electromagnetic heating for cancer treatment. Learning to apply the technique, whether formulating your own routines or using a commercial code, will yield insight for design. The method can produce time-domain animated displays of the ﬁelds that show radiation centers and where the ﬁelds propagate, but the user must learn to interpret these new displays. It will be worth your effort to learn this task. The time-domain responses using impulse signals can produce solutions over a wide band of frequencies when converted to the frequency domain using the discrete Fourier transform (DFT). The only drawback is the computer run time required. 2-6.1 Implementation By using a direct implementation of Maxwell’s curl equations in the time domain, you do little analytical processing of the equations. No vector potential or Green’s

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FINITE-DIFFERENCE TIME-DOMAIN METHOD

functions are developed as in frequency-domain methods. Although the antenna may be volumetrically complex and contain many different materials, the method yields sparse matrices rather than the dense matrices produced by moment methods. It is a direct solution that does not require the inversion of large matrices and includes only nearest-neighbor interactions. Having only nearest-neighbor interactions means that it is possible to run problems on parallel machines. You need to embed the antenna in a rectangular region and divide it into rectangular cubical cells with sizes ranging from 10 to 20 samples per wavelength at the highest frequency where analysis is desired. The outer surfaces contain absorbing boundaries to eliminate reﬂections that would produce errors. Formulating absorbing boundary conditions has been a signiﬁcant part of the method. You need to locate a solution surface between the absorbing boundaries and the antenna outer surface where we compute currents by using the equivalence theorem. The DFT of the time response determines the radiation pattern at a given frequency after the equivalent currents are found. If you need the pattern amplitude in only a few directions, the time-domain radiation can be found directly: for example, the gain in one direction. We can formulate some problems in one or two dimensions if they possess symmetry instead of the three-dimensional rectangular cube. The solution time is reduced dramatically, and the time-animated presentation may provide sufﬁcient insight when the radiation pattern is found in two dimensions. Because this is a time-domain analysis, we need to excite the structure with a pulse. You use the pulse frequency power response to normalize the patterns and compute gain. When the formulation includes the material losses, the efﬁciency of the antenna can be found since the dissipation in the inner cells prevents the radiation from reaching the outer surface. 2-6.2 Central Difference Derivative Numerical derivatives have greater potential for errors than integrals, but FDTD uses them to reduce Maxwell’s differential curl equations to simple difference equations. A second-order accurate formula for a derivative can be found by using central differences instead of using the difference between the value at a location or time and the value at the next point in a sequence of evenly spaced points: f (u0 + u/2) − f (u0 − u/2) ∂f = + O( u)2 ∂u u

(2-49)

We can use ﬁnite differences to solve the curl equations provided that we use electric and magnetic ﬁelds spaced at half intervals because each is related to the derivatives of the other ﬁeld and we want to use central differences to reduce error. Because Maxwell’s equations involve time derivatives, we need to calculate the electric and magnetic ﬁelds at interspersed half time intervals. 2-6.3 Finite-Difference Maxwell’s Equations Consider Maxwell’s curl equations in the time domain, including lossy materials: ∂H 1 = − (∇×E − M + σ ∗ H) ∂t µ ∂E 1 = − (∇×H − J + σ E) ∂t ε

(2-50)

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RADIATION STRUCTURES AND NUMERICAL METHODS

Equations (2-50) contain the source currents J and M and include losses due to conducting dielectric material σ and magnetic material losses σ ∗ . Both equations have the same form, with only an interchange of symbols. Expanding the curl operator, we get the following equation for the x-component of the magnetic ﬁeld: 1 ∂Ey ∂Ez ∂Hx ∗ = − − Mx − σ Hx (2-51) ∂t µ ∂z ∂y The x-component of the electric ﬁeld has the same form but with the interchanges H → E, E → H , M → J , and σ ∗ → σ . You obtain the equations for the y- and z-components by a cyclic variation (repeating pattern of interchanges) x → y → z → x → y, and so on. For example, the equations are reduced to two dimensions by leaving out the y-component. FDTD calculates the ﬁeld at discrete times and locations on a grid. The ﬁelds can be represented as an indexed function using integers: f (i x, j y, k z, n t) = f (i, j, k, n) Because we use central differences [Eq. (2-49)], for derivatives, and the magnetic (electric) ﬁeld is found from the space derivative of the electric (magnetic) ﬁeld, the magnetic and electric ﬁelds need to be spaced a half-space interval apart. The time derivative becomes f (i, j, k, n + 12 ) − f (i, j, k, n − 12 ) ∂f (i, j, k, n) = ∂t t and means that the electric and magnetic components are interspersed at t/2 times that which produces a leapfrog algorithm. We substitute these ideas into Eq. (2-51) to derive the time-stepping equation for one component: Hx (i − 12 , j, k, n + 1) =

1 − σ ∗ (i − 12 , j, k) t/2µ(i − 12 , j, k) 1 + σ ∗ (i − 12 , j, k) t/2µ(i − 12 , j, k) +

Hx (i − 12 , j, k, n)

t/µ(i − 12 , j, k) 1 + σ ∗ (i − 12 , j, k) t/2µ(i − 12 , j, k)

Ey (i − 12 , j, k + 12 , n + 12 ) − Ey (i − 12 , j, k − 12 , n + 12 ) z −

Ez (i − 12 , j + 12 , k, n + 12 ) − Ez (i − 12 , j − 12 , k, n + 12 ) y

− Mx (i − 12 , j, k, n + 12 )

(2-52)

FDTD uses similar equations for the other components [18,19]. Yee’s Cell Figure 2-15 shows one cubic cell and the components of the ﬁelds. When we consider the upper face, we see that the magnetic ﬁeld components are spaced a

FINITE-DIFFERENCE TIME-DOMAIN METHOD

79

Hy

z Ez Hx

Hx Hz Hy Hz

Ey

Hz Ex Hx (i,j,k)

Hy y

x

FIGURE 2-15 Unit cell of a Yee space lattice showing time and space separation of electric and magnetic ﬁelds in a cell. (From [15], Fig. 1,  1966 IEEE.)

half space interval from the central electrical ﬁeld and the arrows show the direction of ﬁelds. Although it would appear that the electric ﬁeld is different on the upper and lower face along the z-axis, the method assumes that the ﬁeld is constant throughout the cell. The magnetic ﬁelds shown are at the center of adjoining cells. A leapfrog solution uses stored values of the electric ﬁelds to calculate the magnetic ﬁelds at a half time interval later and stores these values. In the second step the solution takes another half time step and uses the stored values of the magnetic ﬁelds to calculate the electric ﬁelds. The method gains stability by using the half time steps and by solving for both electric and magnetic ﬁelds. Although the ﬁelds are a half time step out of synch, we can average between the two half time steps to produce simultaneous ﬁelds at a point, but we only need to do this when calculating equivalent currents on the surface used for far-ﬁeld pattern calculations. 2-6.4 Time Step for Stability You need to pick the time step to produce a stable solution. Consider a plane wave traveling through the cubes. If the time step is too large, the wave can pass through more than one cell for each time step. At that point the solution cannot follow the actual wave propagation and fails. We must reduce the time step until it is less than the Courant condition or the wave propagation rate. Consider the fastest-moving wave in the problem, usually free space, and for equal sides to the cube, we compute the time step from the velocity and cell length: x v t ≤ √ d

(2-53)

The cell length is x and the number of dimensions is d. The time step must be lower for conducting materials (σ > 0) to produce a stable solution. The magic step uses

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RADIATION STRUCTURES AND NUMERICAL METHODS

the equality and produces the most stable solutions. If you pick unequal sides to the rectangular cell, Eq. (2-53) is modiﬁed. 2-6.5 Numerical Dispersion and Stability FDTD analyses produce solutions that fail to propagate through the cells at the proper phase velocity in all directions. The propagation velocity depends on the cell size in wavelengths; it has a frequency-dependent component. You need to consider this numerical dispersion because it affects accuracy. Because the waves travel at different velocities in different directions, the dispersion problem increases for large structures where many time steps must be taken. After many steps, signals disperse because they have taken different routes and fail to add together in the correct phase. Finer cells solve the problem, but the computation requirements grow rapidly. The equation for the propagation constant can be found from considering the FDTD formulation to produce the following equation for three-dimensional problems: ky y 2 kz z 2 1 ω t 2 kx x 2 1 1 1 = + + sin sin sin sin c t 2 x 2 y 2 z 2 (2-54) The factor kx is the FDTD propagation constant in the cells along the x-axis, only approximately the same as kx , the actual propagation constant in the structure. The yand z-axes have similar problems. If you take the limit as cell length approaches zero, u → 0, and so on, then sin(au)/u → a. Because t → 0 as the cell size shrinks for the solution still to satisfy the Courant limit, Eq. (2-54) reduces to the expression ω 2 = kx2 + ky2 + kz2 (2-55) c Equation (2-55) is the normal propagation constant equation for a plane wave in space and shows that the cell propagation constants converge to the correct values as the cell size shrinks. If you formulate a problem in one or two dimensions, you remove terms from the right side of Eq. (2-54) to determine the dispersion relationship. Absorbing boundary conditions (ABCs) can cause numerical instabilities. ABCs approximate inﬁnite space to simulate radiation by the antenna into space. FDTD problems must be placed in a ﬁnite number of cells because each cell requires computer storage. Every FDTD problem uses a ﬁnite number of cells for the ABCs with more cells required in the directions of maximum radiation. ABCs degrade as the number of time steps increases and eventually leads to numerical instabilities. A lively research on ABCs has produced good ones, but be aware that most have been found to produce problems at some point. If you write your own analyses, you will need to ﬁnd appropriate ones. Commercial codes will give their limitations. At one time, ABCs limited solution dynamic range, but ABCs are now available that produce reﬂection coefﬁcients from 10−4 to 10−6 . Numerical dispersion limits the dynamic range as well. Remember that the antenna will be modeled with small cubes that limit the resolution of the results. The errors of modeling lead to solution errors that limit the dynamic range. 2-6.6 Computer Storage and Execution Times The antenna to be analyzed is modeled by a set of cubic cells. Choosing an appropriate number is an art. Similarly, it will be necessary to have a meshing program. Using a

FINITE-DIFFERENCE TIME-DOMAIN METHOD

81

two-dimensional model will greatly reduce computer storage and run time. Remember that our purpose should be to gain insight unavailable from measurements. The calculations require the storage of three components of both the electric and magnetic ﬁelds in each Yee cell. Because we solve the problem in the time domain, the components are only real numbers, unlike frequency responses, which use complex numbers for each component. The material properties of the cells can be indicated with short 1byte integers provided that there are no more than 256 different ones. Single-precision storage of the components requires 30 bytes for each cell; double-precision storage requires 54 bytes. A three-dimensional problem with 200 cells on a side contains 8 M cells and would need 240 Mbytes of storage for single-precision and 432 Mbytes for double-precision components. At each time step approximately 10 ﬂoating-point operations (ﬂops) are needed for each component in each cell. We must run the time steps until the input pulse has peaked and died out in each cell. This takes about 10 times the number of cells in the longest direction (maximum number along one axis). The three-dimensional problem with 200 cells on a side runs for 2000 time steps and requires 60 ﬂops times the number of cells. The solution needs 2000 × 8 M × 60 ﬂops = 960 Gﬂops for completion. 2-6.7 Excitation We specify the excitation of an antenna in the time domain since FDTD operates in the time domain. If all we need is a single-frequency solution, a ramped sinusoidal waveform can be applied. The waveform is tapered from zero in about three cycles and the FDTD solution steps continue until a steady state is reached. It is more efﬁcient to use a waveform that gives a wide-frequency-range response after performing a discrete Fourier transform on the radiating boundary to compute equivalent currents used at a given frequency. The computer storage and run times are the same for the wideband response as the single-frequency response. A suitable wide-bandwidth excitation is the differentiated Gaussian pulse shown in Figure 2-16: (t/τp )2 − 1 t (2-56) Vinc (t) = −V0 exp − τp 2 We calculate the frequency response of the differentiated Gaussian pulse from the Fourier transform of Eq. (2-56): √ (ωτp )2 − 1 (2-57) Vinc (ω) = −j ω 2π τp2 V0 exp − 2 The spectrum of Eq. (2-57) peaks for ωp = 1/τp . Figure 2-17 gives the normalized frequency response and shows that the −20-dB-level normalized frequency extends from 0.06 to 2.75. For example, if we wanted to center the frequency response at 10 GHz, the normalizing pulse time is easily found: τp =

1 = 1.592 × 10−11 s = 15.92 ps 2π(10 × 109 )

A check of Figure 2-17 shows that the antenna frequency response could be found from 2 to 22 GHz with only a 10-dB loss in dynamic range compared to the response at 10 GHz. A single time response computation yields a wide-frequency-range response.

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Normalized Voltage

Gaussian Pulse

Differentiated Gaussian Pulse

Normalized Time

Amplitude, dB

FIGURE 2-16 Differentiated Gaussian pulse time response used in FDTD analysis.

Gaussian Pulse Differentiated Gaussian Pulse

Normalized Frequency

FIGURE 2-17

Differentiated Gaussian pulse normalized frequency response.

A sinusoidal modulated Gaussian pulse produces a narrow-bandwidth excitation useful in visualization because the bandwidth of the pulse can be controlled:

(t/τp )2 Vinc (t) = V0 exp − sin ω0 t 2

(2-58)

FINITE-DIFFERENCE TIME-DOMAIN METHOD

83

The unmodulated Gaussian pulse shown in Figure 2-16 has a low-pass frequency response: √ (ωτp )2 Vinc (ω) = 2πτp V0 exp − (2-59) 2 Figure 2-17 gives the low-pass frequency response of the Gaussian pulse with a −4.37dB response at ωp = 1/τp . The sinusoidal modulation centers the frequency response of the Gaussian pulse around ω0 , and the convolution of the two frequency responses produces a two-sided response of the Gaussian pulse.

2-6.8 Waveguide Horn Example [19] The literature contains solutions for the patterns of a number of antennas. Figure 2-18 shows the meshing of a commercial standard gain horn analyzed and compared to measurement. The horn operates from 8.2 to 12.4 GHz. The horn has a radiating aperture that is 110 mm wide and 79 mm high and a bell length of 228 mm. The 51-mm length of the input waveguide and the details of the feed probe were included in the model. Placing a perfectly magnetic conductor through the midsection of the horn uses symmetry to halve the number of cells to a uniform mesh of 519 × 116 × 183 Yee cells. Ten cells were used on the outside for the ABCs around the sides of the horn and 40 cells for the front ABCs in the maximum radiation direction. The model placed 20 cells between the edge of the horn and the equivalent current surface used for pattern calculations. The longest side of the grid determined the number of time steps at 10 times the number of cells = 5190 time steps. The model contains approximately 11 M Yee cells that require 330 Mbytes of computer storage. Assuming that the problem

Probe

WR-90 Waveguide

FDTD Cells Shown 2X

Coaxial Line a

b a

lw D b b

a

FIGURE 2-18 FDTD model of a standard gain horn. (From [17], Fig. 7.17,  1998 Artech House, Inc.)

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RADIATION STRUCTURES AND NUMERICAL METHODS

(a)

(b)

FIGURE 2-19 FDTD calculated electric ﬁeld in the vertical symmetrical plane of a standard gain horn: (a) early time with a pulse in the throat; (b) pulse leaving the mouth of the horn. (From [17], Fig. 7.20,  1998 Artech House, Inc.)

takes 60 ﬂops per cell for each time step, the solution required 3.43 Tﬂops of computer calculations. The initial calculation used a differentiated Gaussian pulse excitation with τp = 15.9 ps that centered the response at 10 GHz. The calculation produced patterns that matched measurements. A second calculation used a sinusoidal modulated Gaussian pulse with the time constant 79.6 ps. This pulse time constant gives a normalized frequency of 2 GHz for the Gaussian pulse. The −3-dB frequency is 0.83 times the normalizing frequency. The pulse is centered at 10 GHz with a 3-dB bandwidth of 3.32 GHz. Figure 2-19 shows the ﬁelds when the pulse reached the horn aperture. Note the high ﬁelds in front of the horn and the amount of ﬁelds still radiating beyond and behind the aperture. By using a sinusoidal modulated pulse, the visual display contains nulls that improve its clarity. 2-7 RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION Ray optics can give you a good physical feel for radiation and spur design ideas, but we need to question the accuracy of their use. Ray optics or geometric optics (GO) methods come from the design of lens and optical reﬂectors where the wavelength is very short compared to the size of the object being analyzed, whereas we may be interested in analyzing or designing an antenna on a structure only a few wavelengths in size. Below we show that GO is essentially correct over most of the radiation sphere and that by using elements of the geometric theory of diffraction [GTD (UTD)], the pattern prediction can be improved. In this case improvement means that we will increase the area of the radiation pattern that becomes more accurate. You will discover that it takes

RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

85

an increasing amount of effort to improve small areas of the pattern prediction, and at some point you should decide that it fails to give enough improvement to justify the work. Your real design purpose is to determine antenna dimensions that produce the desired antenna response. Of course, as the expense of the antenna increases, your customer may demand better predictions of the ﬁnal result, and then the cost of a better analysis is justiﬁed. You need to accept a new approach. Even though a part of the pattern prediction shows errors, obvious discontinuities, it only means that the pattern is inaccurate in directions near them and that over most of the radiation sphere the prediction is essentially correct. Discussion of this method begins with simple examples given in two-dimensional space that introduce the ideas behind GO and GTD. These examples can ignore the details of rotation of polarization directions because the waves are either polarized with the electric ﬁeld normal to the page or located in the plane of the page. We consider radiation blockage by objects, the reﬂection of rays by the objects, and the diffraction of rays around edges that ﬁlls in the pattern in the shadow regions and across the boundary of the last reﬂected ray. After the discussion of simple examples, the key points of GTD will be given for use in three-dimensional problems. This involves the rotation of coordinate systems so that ray polarizations line up with planes of incidence for reﬂections, with edges for diffraction and curvature directions on curved surfaces that shed rays around the object into the shadow. You will need to investigate the references if you want to develop your own routines, but this discussion will introduce you to the topic and give you an appreciation of the method so that you can use available computer programs and understand their limitations. GO uses ray methods to approximate electromagnetics. It is exact only in the limit of zero wavelength (inﬁnite frequency), but we gain useful insight from it at any frequency. It will not give good results close to physical boundaries; but when we include the GTD, the results are accurate down to one-wavelength sizes and are useful at λ/4 sizes. GO gives us physical insight when we deal with reﬂectors. We must consider three aspects to use GO fully: (1) ray reﬂections, (2) polarization, and (3) amplitude variations along the ray path and through reﬂections. 2-7.1 Fermat’s Principle Rays travel through a medium at the speed of light determined by the index of refrac √ tion: n = εr µr . We deﬁne the optical path length as C n dl, where C is a prescribed path in space. Fermat’s principle determines the paths of rays between two points. It states that the optical path length is stationary along a valid ray path. An expression is stationary when its ﬁrst derivatives are zero and the optical path is a minimum (or maximum). We use Fermat’s principle to trace ray paths through reﬂection or refraction by searching for the minimum optical path lengths. We can ﬁnd more than one possible ray between points because Fermat’s principle requires only a local minimum. When we exclude the boundaries of lenses, regions of homogeneous medium, rays travel in straight lines. 2-7.2 H -Plane Pattern of a Dipole Located Over a Finite Strip Figure 2-20 illustrates the geometry of this problem and the various regions of the analysis. The diagram shows the end of the dipole rod with the two rods located

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RADIATION STRUCTURES AND NUMERICAL METHODS

I RB

RB

l/4 II

II 3l/8

5l/8 SB

SB III

FIGURE 2-20 GTD example using a two-dimensional model of a dipole located over an asymmetrical ground plane.

normal to the page. The dipole pattern is omnidirectional in the page with the electric ﬁeld directed normal to the page. When we trace rays from the dipole to the ﬁnite strip, we discover two signiﬁcant directions on both sides of the strip. The dashed boundaries labeled RB (reﬂection boundary) are the directions of the last rays reﬂected from the strip. Similarly, the dashed boundaries labeled SB (shadow boundary) are the last rays of radiation not blocked by the strip. The radiation in region I results from the sum of the direct radiation from the dipole plus the radiation reﬂected by the strip. Only direction radiation from the dipole occurs in the two parts of region II. Finally, region III is totally blocked from any radiation by a direct or reﬂected ray. This region receives rays diffracted around the edges. If we add the direct and reﬂected rays in an analysis, we obtain the pattern given in Figure 2-21, which also traces the actual pattern. The pattern, using only the direct and reﬂected rays, accounts for the phasing between the direct radiation from the dipole and an image dipole located below the strip. If you compare the two traces on Figure 2-21, you see that the two patterns are similar near θ = 0, but the direct plus reﬂected ray pattern has discontinuities at the SBs and RBs. Figure 2-22 gives the results for the same analysis, but using a 5λ-wide ground-plane strip. When using the larger strip, the two patterns match to about 80◦ , and in the second case the simple analysis is correct over most of the forward semicircle. Simple geometric optics gives good results for large objects provided that you realize the patterns will contain discontinuities. Removing the discontinuities requires extra effort. A discontinuity in the pattern cannot exist because shadow and reﬂection boundaries occur in free space. It takes a material boundary to produce a discontinuous ﬁeld. But, for example, the tangential electric ﬁeld must be continuous across even material boundaries. Edge diffraction solves the discontinuity problem. Figure 2-23 gives the pattern of the edge diffraction for both edges normalized to the total pattern. The edge diffraction has matching discontinuities to the sum of the direct and reﬂected rays at the SBs and RBs. The UTD (uniform theory of diffraction) technique [20, p. 55] calculated these diffractions. When these diffractions are added to the direct and reﬂected ray radiation, the total pattern given in Figure 2-21 is obtained. The dipole, its image in the ground plane, and the two edge diffractions form a four-element array where each element has a unique pattern. Adding edge diffractions to the geometric optics ﬁelds removes the discontinuities and allows calculation of the pattern behind the strip ground plane.

RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

87

Total Pattern

Direct + Reflected

FIGURE 2-21 H -plane pattern of a dipole over asymmetrical ground using direct and reﬂected rays compared only to a full solution for the 1λ ground plane of Figure 2-20.

Total

Direct + Reflected

FIGURE 2-22 H -plane pattern of a dipole over symmetrical ground using direct and reﬂected rays compared only to a full solution for a 5λ ground plane.

2-7.3 E -Plane Pattern of a Rectangular Horn Figure 2-24 illustrates the cross section of a horn or, in this case, a two-dimensional approximation to a horn. The waveguide feeds the horn and produces a uniform aperture distribution in the E-plane. In this model the direct GO radiation is a constant wedge signal as shown in Figure 2-25 ranging between −15◦ and +15◦ . The reﬂected pattern

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RADIATION STRUCTURES AND NUMERICAL METHODS

FIGURE 2-23 GTD edge diffraction of an H -plane pattern for an asymmetrical 1λ ground plane under a dipole.

combines with the direct radiation and produces the same pattern. Figure 2-25 also shows the diffraction patterns from the two edges. These peak along the plates and exhibit a discontinuity at the same angle as the GO ﬁeld. Each diffraction pattern has a discontinuity on one side at 90◦ because the mouth of the horn blocks the diffraction from the opposite edge. When we add the diffracted ﬁelds to the GO ﬁeld, the pattern shown in Figure 2-26 is obtained. By just adding the three components, we obtain an accurate pattern of the horn over most of the angles of the plot. At 90◦ we see discontinuities in the pattern caused by not considering enough terms in the GTD calculation. You need to realize that these discontinuities only cause pattern errors at nearby angles. The majority of the pattern is correct. We need another term to correct the pattern near 90◦ . The blockage of the diffraction from one edge by the mouth of the horn causes a secondary diffraction at that edge. We call this double diffraction. Some available programs do not implement double diffraction because the general three-dimensional double diffraction takes considerable calculation due to the extensive ray tracing required. In these cases you must accept the pattern discontinuities. Some programs calculate double diffraction as an option, but turning on this option will slow the calculations. Figure 2-27 gives the pattern when double diffraction is included. Double diffraction reduces the discontinuity at 90◦ , but a small discontinuity remains. Adding triple diffraction would reduce this further, but the pattern area affected by the small discontinuity has shrunk. A new discontinuity near 60◦ appeared in the pattern after adding double diffraction at the

RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

89

FIGURE 2-24 Geometry of a two-dimensional model of a rectangular horn used for GTD analysis.

mouth of the horn. We could continue to add another term to remove this one or just accept it. 2-7.4 H -Plane Pattern of a Rectangular Horn The tangential electric ﬁelds vanish at the walls of the two-dimensional horn in the H -plane. This affects the GO ﬁeld and produces the following equation for them: E GO = cos

π tan θ e−j kR √ 2 tan α R

(2-60)

Equation (2-60) includes the phasing term and square-root spreading factor of a twodimensional ﬁeld. The horn walls tilt from the centerline by the angle α. Figure 2-28 plots the GO ﬁeld and shows that it vanishes at the walls. We do not expect edge diffraction because the ﬁeld vanishes at the edges, but Figure 2-28 shows diffraction patterns that peak in the direction of the walls. We call this new term slope diffraction. This new type requires another set of coefﬁcients not identical to the edge (or wedge) diffraction coefﬁcients. While the amplitude of the edge diffraction is proportion to the ﬁeld incident on the edge, the amplitude of slope diffraction is proportional to the derivative of the ﬁeld in the direction normal

90

RADIATION STRUCTURES AND NUMERICAL METHODS

FIGURE 2-25 E-plane pattern of a rectangular horn with a GO term (solid curve) and edge diffractions (dashed curves).

FIGURE 2-26 Combination of GO and edge diffractions in the E-plane pattern of a rectangular horn.

to the edge. We ﬁgure the same geometric factors for both edge and slope diffraction but now must calculate the normal derivative of the incident electric ﬁeld. Figure 2-29 plots the H -plane pattern of the horn. The pattern fails to predict a pattern behind it. The E-plane diffraction produces a back hemisphere pattern for a real horn, but our two-dimensional model does not include the E-plane. 2-7.5 Amplitude Variations Along a Ray Power decreases in a general ray as the distance from the source increases. If we expand the constant-phase surface (eikonal) about the ray in a Taylor series, we obtain a surface described by its radii of curvature [20, p. 55]. The maximum and minimum

RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

91

FIGURE 2-27 E-plane pattern of a rectangular horn combining GTD terms of direct GO, edge diffractions, and double diffractions between edges.

FIGURE 2-28 H -plane pattern of a rectangular horn with a GO term (solid curve) and edge slope diffractions (dashed curves).

values lie in the orthogonal principal planes. These radii of curvature determine the amplitude spread of the wave from point to point on the ray. We compute the ratio of differential areas about the ray at two locations as dA2 ρ1 ρ2 = dA1 (ρ1 + d)(ρ2 + d)

(2-61)

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RADIATION STRUCTURES AND NUMERICAL METHODS

FIGURE 2-29 H -plane pattern of a rectangular horn by GTD analysis by combining direct GO ﬁeld and edge slope diffraction.

r1 dA1 r2

dA2

d

Principal Planes S

FIGURE 2-30 Astigmatic ray.

where ρ1 and ρ2 are the principal radii of curvature and d is the distance between two points on the ray (Figure 2-30). The electric ﬁeld variation along the ray becomes E0 e

−j kd

ρ1 ρ2 (ρ1 + d)(ρ2 + d)

(2-62)

for the astigmatic ray spreading from unequal radii of curvature. When d = −ρ1 or d = −ρ2 , GO fails because it predicts an inﬁnite power density. We call these locations caustics. Remember that the ray always has differential area and never has any real area as implied by Figure 2-30. We have three special cases of the astigmatic ray: 1. Spherical wave, ρ1 = ρ: E0 e−j kd

ρ ρ+d

(2-63)

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RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

2. Cylindrical wave, ρ1 = ∞: E0 e−j kd 3. Plane wave, ρ1 = ρ2 = ∞:

ρ ρ+d

E0 e−j kd

(2-64)

(2-65)

The plane wave does not spread but has constant amplitude as distance changes. Both cylindrical and plane waves require inﬁnite power, and they are therefore nonphysical, but we ﬁnd them convenient mathematically. 2-7.6 Extra Phase Shift Through Caustics We cannot determine the ray amplitude at a caustic but can determine its amplitude and phase on either side. Passage through a caustic causes an extra phase shift to the ray [21, p. 31]. The denominator factors in the square root of Eq. (2-62) produce a 180◦ sign change when the ray distance factor d passes through either ρ1 or ρ2 . The square root changes 180◦ to +90◦ (ej π/2 ) or −90◦ (e−j π/2 ), depending on the direction of movement along the ray. When tracing a ray moving through a caustic in the direction of propagation, you multiply by ej π/2 . The ﬁeld is multiplied by e−j π/2 for a ray traced in the opposite direction of propagation. 2-7.7 Snell’s Laws and Reﬂection We derive Snell’s laws of reﬂection and refraction from Fermat’s principle. The two laws of reﬂection are given as: 1. The incident ray, the reﬂected ray, and the normal of the reﬂecting surface at the point of reﬂection lie in the same plane. 2. The incident and reﬂected rays make equal angles with the surface normal. Implicit in Snell’s laws is the idea that locally the wavefront behaves like a plane wave and that the reﬂector can be treated as a plane. Given the direction of the incident ray S1 , reﬂected ray S2 , and the reﬂector normal n, Snell’s laws of reﬂection can be expressed vectorially [22]: n×(S2 − S1 ) = 0

n·(S1 + S1 ) = 0

(2-66)

We combine Eq. (2-66) to determine the ray directions before or after reﬂection: S1 = S2 − 2(S2 · n)n

S2 = S1 − 2(S1 · n)n

(2-67)

Snell’s law of refraction can also be expressed vectorially as n×(n2 S2 − n1 S1 ) = 0 where n1 and n2 are the index of refractions in the two mediums.

(2-68)

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RADIATION STRUCTURES AND NUMERICAL METHODS

2-7.8 Polarization Effects in Reﬂections The electric ﬁeld is orthogonal to the ray direction (a free-space wave) and is described by a two-dimensional polarization space (Section 1-11). We can describe polarization in any conveniently rotated two-dimensional basis vectors in the plane with the ray vector as its normal. We will use a ray-ﬁxed coordinate system that changes direction after a reﬂection: (2-69) Ei = ai|| Ei|| + ai⊥ Ei⊥ where ai|| is a unit vector in the plane of incidence and ai⊥ is perpendicular to the plane of incidence. We compute ai⊥ from the normal to the plane n at the reﬂection point and the incident ray unit vector Si : ai⊥ =

Si × n |Si × n|

ai||

ai⊥

=

(2-70)

× Si

After reﬂection, we calculate the output ray-ﬁxed polarization vectors using the output ray Sr : ar⊥ = ai⊥ and ar|| = ar⊥ × Sr Ei|| is the incident electric ﬁeld in the direction of ai|| and Ei⊥ is in the direction of ai⊥ . Of course, the direction of unit vector a|| changes from incident to reﬂected rays. The electric ﬁeld parallel to the reﬂector surface must vanish on the conductor surface: Er⊥ = −Ei⊥

(2-71)

where Er⊥ is the reﬂected ﬁeld along ai⊥ . We calculate the reﬂection properties of E|| from the corresponding magnetic ﬁelds parallel to the surface: Hr|| = Hi||

(2-72)

By combining Eqs. (2-71) and (2-72), we obtain the dyadic relation for the ray-ﬁxed coordinate system: Ei|| 1 0 Er|| = (2-73) 0 −1 Er⊥ Ei⊥ where Er|| and Ei⊥ are the reﬂected ﬁeld components. At each reﬂection we rotate the polarizations to align ai⊥ with the normal to the plane of incidence. We can express Eq. (2-73) as a dyadic in terms of the incident and reﬂected wave polarization vectors = ai ar − ai ar . Of course, the alternative method is to describe polarizations in a R || || ⊥ ⊥ ﬁxed three-dimensional coordinate system, but it requires a 3 × 3 reﬂection matrix. 2-7.9 Reﬂection from a Curved Surface A wave reﬂected from a curved surface changes its radii of curvature and principal planes. The ﬁeld along the reﬂected ray is given by ρ1r ρ2r Er (s) = Ei0 ·R e−j ks (2-74) r (ρ1 + s)(ρ2r + s)

RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

95

where s is the distance along the ray from the reﬂection, ρ1 and ρ2 the reﬂected ray the reﬂection dyadic. Ei0 the incident ray electric ﬁeld. For radii of curvature, and R a ﬂat surface we use images of the incident ray caustics for ρ1r and ρ2r , but in general, ρ1r and ρ2r become 1 1 = ρ1r 2

1 1 + i ρ1i ρ2

+

1 1 = ρ2r 2

1 f1

1 1 + i ρ1i ρ2

+

1 f2

(2-75)

where f1 and f2 are generalized focal lengths of the surface. The spreading factor of Eq. (2-74) simpliﬁes in the far ﬁeld:

ρ1r ρ2r ≈ r (ρ1 + s)(ρ2r + s)

r r ρ1 ρ2 s

Kouyoumjian and Pathak [23] derived formulas for the focal lengths of a surface. We start with a surface with principal radii of curvature R1 and R2 with directions u1 and u2 at the point of reﬂection. For an incident ray with principal axes deﬁned by unit vectors xi1 and xi2 , we deﬁne a matrix relation between the incident ray and surface principal curvature directions:

xi · u θ = 1i 1 x2 · u1

xi1 · u2 xi2 · u2

(2-76)

where the determinant is |θ | = (xi1 · u1 )(xi2 · u2 ) − (xi2 · u1 )(xi1 · u2 ). Given the angle of incidence θ i , the following are the focal lengths: 2 2 2 2 θ21 + θ12 + θ11 1 cos θ i θ22 = + f1,2 |θ |2 R1 R2 2 2 2 2 2 θ21 − θ12 − θ11 1 1 1 4 cos θ i θ22 1 1 − i + − i + ± 2 |θ |2 R1 R2 ρ1i ρ2 ρ1i ρ2

1/2 2 2 2 2 2 θ21 + θ12 + θ11 θ22 4 cos2 θ i 4|θ |2 + + − (2-77) |θ |4 R1 R2 R1 R2 With a single reﬂection, we need not compute the direction of the principal axes. We need only the focal lengths. Multiple reﬂections require knowledge of the reﬂected-ray principal plane directions. Deﬁne the following matrices to determine the directions of the principal axes after reﬂection: 

1  ρ1i Qi0 =   0

 0   1  ρ2i

 1  C0 =  R1 0

 0

 1  R2

Qr = Qi0 + 2(θ −1 )T C0 θ −1 cos θi br1 = xi1 − 2(n · xi1 )n

br2 = xi2 − 2(n · xi2 )n

96

RADIATION STRUCTURES AND NUMERICAL METHODS

where n is the surface normal at the reﬂection point. One principal-axis direction is xr1

r Q22 − 1/ρ1r br1 − Qr12 br2 = 2 Qr22 − 1/ρ1r + (Qr12 )2

(2-78)

We derive the other from the cross product of Eq. (2-78) and the reﬂected ray unit vector: xr2 = −Sr × xr1 (2-79) We must reapply Eqs. (2-75) through (2-79) for every reﬂection. We use Eqs. (2-75) through (2-79) for analysis, but except for computer optimizations, they cannot be applied directly to synthesis. If we limit the reﬂectors to ﬁgures of rotation, the radii of curvature are given by the meridians and parallels and these problems reduce to two dimensions. Similarly, a cylindrical reﬂector fed with a cylindrical wave [Eq. (2-64)] reduces the problem to two dimensions. The incident and reﬂected waves remain in the single plane chosen for the reﬂector analysis. 2-7.10 Ray Tracing Tracing rays through a reﬂector system is conceptually straightforward. Where a ray strikes a reﬂector, we compute the normal to the surface. By using Eq. (2-67), we solve for the reﬂected-ray direction. Equation (2-73) determines the polarization effects when we express the incident and reﬂected rays in the ray-ﬁxed coordinates. We use geometric arguments to determine the amplitude variation along the ray through the reﬂection instead of the general expressions given above. We experience difﬁculty when we try to discover the reﬂection points for given ﬁeld and source points. No analytical expressions exist for calculating the reﬂection point of a general surface. The usual computer routines search for the minimum optical path length (Fermat’s principle) without using Eq. (2-67), since a local minimum will satisfy this equation. 2-7.11 Edge Diffraction Keller [24] extended the idea of reﬂection to edge diffraction by applying a generalized Fermat’s principle to the rays. Figure 2-31 illustrates the rays in edge diffraction and the associated polarization directions. The ﬁgure shows the edge vector at the diffraction point. The vector cross product between the edge vector and the incident ray points in the direction of the incident plane normal. We measure the diffraction angle of incidence in this plane between the incident ray and the edge normal. Because diffraction obeys a generalized Fermat’s principle, the diffracted ray exits at the same angle, similar to the reﬂected ray angles. The diffracted rays lie in a cone with the edge vector as its axis. The diffracted rays spread the incident power into a cone. Figure 2-31 shows a particular diffracted ray and how we determine the diffracted ray exiting plane. We deﬁne diffracted ray polarization in terms of the incident and diffracted planes. The vectors are parallel and perpendicular to the two planes. Given the edge unit vector e, you compute the incident perpendicular polarization vector: aφ =

e × S sin β0

(2-80)

RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

97

e^

s^

Efd

Ed

Plane of ^ ^ Diffraction (s, e)

b0

QE

E if′

Edge-Fixed Plane of Incidence (s^′, e^)

E ib′0

b0

^ s′

Edge (a)

f

^

f

^

f′

^

n

Re f Bo lectio und n ary

f′ O

QE w ado y Sh undar Bo

Plane ⊥ e^ at QE

np

(b)

FIGURE 2-31 Ray-ﬁxed coordinates related to edge-ﬁxed coordinates at the edge diffraction point on a curved edge by showing planes of incidence and diffraction. (From [25], Fig. 5,  1974 IEEE.)

where S is the incident ray and β0 is the angle between the edge tangent and the incident ray. The diffracted ray perpendicular polarization is similar to the incident ray aφ = −

e×S sin β0

(2-81)

where S is the diffracted ray unit vector. We have the following vector relations for diffraction: |e × S| = |e × S | and e · S = e · S (2-82)

98

RADIATION STRUCTURES AND NUMERICAL METHODS

We determine the parallel polarization vector along the ray-ﬁxed coordinates by the following cross products: aφ × aβ0 = S

and aφ × aβ0 = S

(2-83)

By using ray-ﬁxed coordinates, the diffraction matrix reduces to 2 × 2. When β0 = π/2, the parallel polarization components are parallel to the edge and the electric ﬁeld vanishes: Eβ0 + Eβ0 = 0. Acoustics calls this the soft boundary condition (Dirichlet); it operates on the parallel polarization components. The perpendicular components satisfy the hard boundary condition (Neumann). At a diffraction point Qe we describe diffraction by the matrix equation

i d Eβ (Qe ) Eβ0 (s) ρ −Ds 0 0 e−j ks (2-84) = i 0 −Dh Eφd (s) s(s + ρ) Eφ (Qe ) where s is the distance from the diffraction point. Diffraction locates one caustic on the diffraction point. We compute the second caustic distance ρ from the incident ray radius of curvature in the plane of incidence ρei and the edge curvature unit vector nˆ e : 1 nˆ e · (ˆs − sˆ ) 1 = i − ρ ρe a sin2 β0

(2-85)

where a is the edge radius of curvature. When a → ∞ (straight edge), the second term of Eq. (2-85) vanishes. A number of factors determine the wedge diffraction coefﬁcients. The diffracting edge factors include (1) the angle between the faces, (2) the edge curvature, and (3) the curvature of the faces. The ray angle factors are (1) the incident angle relative to the edge tangent, (2) the diffraction angle to the shadow boundary, and (3) the angle to the reﬂection boundary. The diffraction coefﬁcients peak at the shadow and reﬂection boundaries. UTD formulation uses characteristic lengths associated with incident and diffracted ray caustics. These many factors are beyond the current discussion. 2-7.12 Slope Diffraction The spatial rate of change of the ﬁeld normal to the edge produces slope diffraction, an added ﬁeld component. This ray optics term also satisﬁes the generalized Fermat’s principle with geometry determined by Eqs. (2-80) through (2-83), and (2-85). The slope diffraction equation has the same form as Eq. (2-84):

d i Eβ (Qe ) Eβ0 (s) ρ −es 0 0 = (2-86) e−j ks 0 −eh Eφd (s) s(s + ρ) Eφi (Qe ) where the diffraction coefﬁcients es,h are related to the ﬁeld derivative normal to the surface: 1 ∂Ds,h ∂ es,h = (2-87) j k sin β0 ∂φ ∂n The term ∂/∂n of Eq. (2-87) indicates the derivative of the incident ﬁelds given in the vector of Eq. (2-86). Equation (2-87) has the term ∂Ds,h /dφ for the soft and hard slope diffraction terms returned from a subroutine; it is only a notational derivative.

RAY OPTICS AND THE GEOMETRIC THEORY OF DIFFRACTION

99

2-7.13 Corner Diffraction Every structural discontinuity diffracts waves. We derive edge diffraction from an inﬁnite wedge where the wedge terminations (corners) produce diffracted rays. Recall from Section 2-4.2 that PTD added currents at edges to handle the effect of not having an inﬁnite surface; the formulation for corner diffraction uses equivalent currents to derive these coefﬁcients. We handle edge diffraction from each edge as always. Since each corner arises from two edges, we compute separate corner diffraction for each edge, two terms per corner. Whereas edge diffraction is bound to a cone, corner diffraction radiates in all directions. The edge must be visible from both the source and receive points before corner diffraction contributes. We must include corner diffraction in any three-dimensional problem. As the source and receiver become farther and farther away from the object, corner diffraction contributions dominate over edge diffractions since it is derived from equivalent currents. 2-7.14 Equivalent Currents GTD fails to predict ﬁelds at caustics. In many cases we consider these points unimportant, but for those cases where we need the ﬁelds, equivalent currents provide the answer. We derive equivalent currents from edge diffraction, which then replaces it and we use them instead of edge diffraction for all pattern points. The use of currents reduces the problem to a PO solution and line integrals are required. We relate the incident ﬁelds expressed in the ray ﬁxed to equivalent currents: √ 2j i Eβ Ds 2πk ej π/4 ηk 0 √ 2j i Eφ Dh 2πk ej π/4 M= 0 k I =

(2-88) (2-89)

The soft and hard diffraction coefﬁcients Ds,h depend on the source and receiver positions. Since we calculate the ﬁelds using vector potentials or dyadic Green’s functions, the formulation has no caustics. They are only associated with a geometric optics solution. Equivalent currents allow the calculation of the ﬁelds directly behind a reﬂector near the axis. The GTD solution produces a caustic as all points along the rim “light up” for an axisymmetrical design. PTD uses equivalent currents in a similar but different way to calculate correct ﬁelds in the same region. Equivalent currents derived from the diffraction coefﬁcients produce the entire solution, since the reﬂector blocks the incident ﬁeld. In PO we continue to include the direct ﬁeld and the induced current radiation on the reﬂector, but add the PTD current radiation. Realize that slope diffraction also adds to the equivalent currents. 2-7.15 Diffraction from Curved Surfaces [26, 27] In one analytical approach to surface-wave radiation we postulate waves bound to a surface that radiate only from discontinuities. Surface waves on inﬁnite structure do not radiate but attenuate exponentially away from the surface, because they are bound to it. We can formulate GTD as radiating from discontinuities, and this produces an approach

100

RADIATION STRUCTURES AND NUMERICAL METHODS

for ﬁelds radiated on the shadowed side of a curved body. The continuous discontinuity of the curved surface causes power to be radiated at every point in the shadow region. These waves radiate tangentially from a wave traveling along a geodesic and bound to the surface. Surface waves require a dielectric coating or a corrugated surface to slow and bind the wave to the surface. The surface curvature slows and binds the wave to the surface without the need for a dielectric or corrugated surface coating. The wave that propagates along the surface sheds power in rays tangentially to it. The rays travel along a surface geodesic from the attachment point to the radiation point. The geodesic curve is a minimum distance path on the surface between two points. In differential geometry it has a broader meaning, but for our purpose, the minimum distance deﬁnition will serve. The curved surface diffraction satisﬁes a generalized Fermat’s principle (minimum distance) as do all other terms of GTD. The best approach uses another ray-ﬁxed coordinate along the surface where the vectors are normal and tangential to the surface at both the attachment and radiation (shedding) points. Curved surface diffraction considers three types of problems with different formulations. Two of them start with an antenna mounted on the surface. We either calculate the pattern in the presence of the curved object or calculate the coupling to a second antenna also mounted on the curved object. The third case determines the ﬁeld scattered for a source located off the surface. All three use the ray-ﬁxed coordinates. We start with the surface normal nˆ and the tangent vector ˆt directed along the geodesic path. A vector cross product deﬁnes the third direction of the local coordinate system. We ˆ and the three vectors form a triad: nˆ × bˆ = ˆt. On a general use the surface binormal b, surface all three vectors change direction as the wave moves along the geodesic. We use the term torsion for a path with a changing binormal. A soft dyadic diffraction coefﬁcient is used with ﬁelds aligned with the attachment point binormal and the tangential shedding point binormal. We apply the hard dyadic diffraction coefﬁcient ﬁelds aligned along the normal vectors. No formulas exist for computing the attachment and shedding points on a general curved surface given the source and receive points. We usually start with a known diffraction and ﬁnd other points by incrementing along the curve by small steps.

REFERENCES 1. L. Diaz and T. A. Milligan, Antenna Engineering Using Physical Optics, Artech House, Boston, 1996. 2. B. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961. 3. P. S. Hacker and H. E. Schrank, Range distance requirements for measuring low and ultralow sidelobe antenna patterns, IEEE Transactions on Antennas and Propagation, vol. AP-30, no. 5, September 1982, pp. 956–966. 4. K. Pontoppidan, ed., Technical Description of Grasp 8, Ticra, Copenhagen, 2000 (selfpublished and available at www.ticra.com). 5. J. H. Richmond, A reaction theorem and its application to antenna impedance calculations, IRE Transactions on Antennas and Propagation, vol. AP-9, no. 6, November 1961, pp. 515–520. 6. R. F. Harrington, Field Computation by Moment Methods, Macmillan, New York, 1968; reprinted by IEEE Press, New York, 1993.

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7. J. H. Richmond, Radiation and scattering by thin-wire structures in homogeneous conducting medium, IEEE Transactions on Antennas and Propagation, vol. AP-22, no. 2, March 1974, p. 365 (see also ASAP wire code). 8. N. N. Wang, J. H. Richmond, and M. C. Gilreath, Sinusoidal reactance formulation for radiation from conducting structures, IEEE Transactions on Antennas and Propagation, vol. AP-23, no. 3, May 1975, pp. 376–382. 9. B. M. Kolundzija and A. R. Djordjevic, Electromagnetic Modeling of Composite Metallic and Dielectric Structures, Artech House, Boston, 2002. 10. A. R. Djordjevic et al., AWAS for Windows Version 2.0: Analysis of Wire Antennas and Scatterers, Artech House, Boston, 2002. 11. A. C. Ludwig, Wire grid modeling of surface, IEEE Transactions on Antennas and Propagation, vol. AP-35, no. 9, September 1987, pp. 1045–1048. 12. A. W. Glisson and D. R. Wilton, Simple and efﬁcient numerical methods for problems of electromagnetic radiation and scattering from surfaces, IEEE Transactions on Antennas and Propagation, vol. AP-28, no. 5, September 1980, pp. 563–603. 13. S. M. Rao, D. R. Wilton, and A. W. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Transactions on Antennas and Propagation, vol. AP-30, no. 3, May 1982, pp. 409–418. 14. B. M. Kolundzija et al., WIPL-D: Electromagnetic Modeling of Composite Metallic and Dielectric Structures, Artech House, Boston, 2001. 15. P. -S. Kildal, Foundations of Antennas, Studentlitteratur, Lund, Sweden, 2000. 16. C. Scott, The Spectral Domain Method in Electromagnetics, Artech House, Boston, 1989. 17. K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Transactions on Antennas and Propagation, vol. 14, no. 3, May 1966, pp. 302–307. 18. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC Press, Boca Raton, FL, 1993. 19. J. G. Maloney and G. S. Smith, Modeling of antennas, Chapter 7 in A. Taﬂove, ed., Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, 1998. 20. D. J. Struik, Differential Geometry, Addison-Wesley, Reading, MA, 1950. 21. D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction, Artech House, Boston, 1990. 22. F. S. Holt, in R. E. Collin and F. J. Zucker, eds., Antenna Theory, Part 2, McGraw-Hill, New York, 1969. 23. R. Kouyoumjian and P. Pathak, The dyadic diffraction coefﬁcient for a curved edge, NASA CR-2401, June 1974. 24. J. B. Keller, Geometrical theory of diffraction, Journal of the Optical Society of America, vol. 52, 1962, pp. 116–130. 25. R. G. Kouyoumjian and P. H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proceedings of IEEE, vol. 62, no. 11, November 1974, pp. 1448–1461. 26. P. H. Pathak, W. D. Burnside, and R. J. Marhefka, A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface, IEEE Transactions on Antennas and Propagation, vol. AP-28, no. 5, September 1980. 27. P. H. Pathak, N. Wang, W. D. Burnside, and R. G. Kouyoumjian, A uniform GTD solution for the radiation from sources on a convex surface, IEEE Transactions on Antennas and Propagation, vol. AP-29, no. 4, July 1981.

3 ARRAYS

We begin with arrays of antennas before discussing particular antenna elements to show the relationship between antenna size and shape and the resulting pattern characteristics. We ignore the feed network design initially and assume that the proper array feed distribution will be obtained. At ﬁrst, we assume a distribution of point sources and compute the approximate array pattern. Working with simple models provides insight rather than accuracy, and later we consider element pattern and interaction. In Chapter 12 we discuss feed network design and analysis in the discussion of phased arrays. The chapter begins with a mathematic description of an array and gives various assumptions used to simplify the expressions. We analyze a simple two-element array to gain insight into the radiation phenomenon and how far-ﬁeld patterns can be found with simple arguments. The discussion of a uniformly spaced linear array shows the Fourier series relationship between array layout and the pattern space given in sin(angle) space. The principal idea is that pattern beamwidth shrinks as the array length increases. If we space the elements too far apart, multiple beam peaks or grating lobes form in the pattern, and we show how to control these grating lobes and their relationship to maximum scan angle, array layout, and element spacing. Phased arrays scan the beam by controlling the relative phasing between the elements. We extend the linear array to planar layouts that produce narrow beams in both principal planes. The planar array design is unchanged from the methods for linear arrays, but the grating lobe analysis shows their unique properties, as they sometimes form outside the plane of scan. We can divide the phased array into pieces to form multiple scanning beams, but the beam shape is determined by the segment size and shape used for each beam. By adding amplitude control the phased array can form multiple beams with beamwidths determined by the entire size of the array. Each element in an array receives a portion of the power radiated by the other elements on transmittal, or scatters power into neighboring elements in reception. The Modern Antenna Design, Second Edition, By Thomas A. Milligan Copyright  2005 John Wiley & Sons, Inc.

102

ARRAYS

103

radiation from each antenna excites currents on its neighboring elements that also radiate, and we associate the total pattern with the antenna input. In an array the effective element patterns change due to this scattering. Because of reciprocity, which says that transmit and received patterns are identical, we can analyze the problem either way. This leads to mutual coupling, which we describe and analyze by mutual impedance (admittance, or scattering) matrices. This phenomenon causes the input impedance of the elements to change as we scan the array. The mutual coupling can lead to scan blindness when the feed reﬂection coefﬁcient grows due to mutual coupling, and the array totally reﬂects the signal into the feed network. If we want the exact pattern designed for, we must compensate the feeding coefﬁcients for the mutual coupling. A discussion of array gain gives two methods of calculation. First, the effective area and the associated gain of a planar array cannot exceed its area when we include the extra half-element spacing area provided by the edge elements. When we space the elements so that their individual effective areas no longer overlap, array gain is the element gain multiplied by the number of elements. We can calculate gain by adding up the input power instead of integrating the pattern to compute total radiated power. We relate input power of elements to the self- and mutual resistances to determine gain of linear and planar arrays using realistic elements. The chapter ends with a discussion of three-dimensional arrays using arbitrarily oriented elements. We add this analysis to the simple array formula to handle the polarization of rotated antennas. Related to this problem is the pointing of an antenna on a positioner. We apply rotation matrices to both problems. An array radiates or receives from two or more antennas at the same frequency. To calculate the ﬁeld radiated from arrays we add the electric ﬁelds radiated from each element. The amplitudes and phases of each antenna, determined by the feed network, give us extra degrees of freedom to shape the pattern and design shifts from radiating elements to the feed network. A single antenna radiates an electric ﬁeld with both polarization components: E = Eθ (θ, φ)θˆ + Eφ (θ, φ)φˆ where Eθ and Eφ are the two complex components (amplitude and phase) referred to some point on the antenna. If we move the antenna or the phase reference point, we only change the antenna radiated phase. We assume that the movement is small enough that the radiation approximation can still be used. Given r as the location of the antenna relative to the phase reference point, the added phase component is ej k·r , where k = 2π/λ(sin θ cos φ xˆ + sin θ sin φ yˆ + cos θ zˆ ) and r = x xˆ + y yˆ + z zˆ is the location of the antenna; k · r is the phase distance from the antenna to the reference plane through the reference point and is deﬁned by the radiation (receiving) direction. The electric ﬁeld radiated from the moved antenna becomes ˆ j k·r [Eθ (θ, φ)θˆ + Eφ (θ, φ)φ]e We assume that nearby objects do not alter the patterns in the movement, but we can alter element patterns if necessary. Suppose that we have an array of antennas located at points r 1 , r 2 , and so on. We obtain the total pattern by adding the electric ﬁelds radiated from each: E=

N i=1

ˆ j k·r i [Eθi (θ, φ)θˆ + Eφi (θ, φ)φ]e

(3-1)

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ARRAYS

Bringing the antennas close together will change the patterns of each because every antenna will block the radiation of the others and the distribution of currents on the elements may be changed. The shape of small resonant antennas limits the possible distribution of currents, but the magnitude and phase may be changed due to the coupling. We make various approximations to Eq. (3-1). Changes in the patterns due to nearby antennas are ignored, and isolated element patterns are used. We assume initially a certain amplitude and phase distribution on the elements and ignore the problem of the feed network. Polarization reduces to a single term for equally polarized elements, such as dipoles, slots, or horns. If the antennas have identical element patterns, we can separate Eq. (3-1) into a product. ˆ E = [Eθ (θ, φ)θˆ + Eφ (θ, φ)φ] Ei ej k·r i (3-2) where Eθ and Eφ are the normalized patterns of the single element. Ei is the electric ﬁeld of the ith element, including the amplitude and phase of the feed distribution. Equation (3-2) describes pattern multiplication that separates the pattern into an element pattern and an array factor. The method requires that all antennas have the same pattern and be orientated in the same direction. The array factor represents the pattern from an array of isotropic pattern antennas. Because array factors can be calculated by hand, we ﬁnd them useful for gaining insight. We leave calculations using Eqs. (31) and (3-2) to the computer. The element patterns themselves could be arrays and we could use pattern multiplication to synthesize planar and volumetric arrays from linear arrays.

3-1 TWO-ELEMENT ARRAY Consider two elements lying on the z-axis and spaced a distance d centered on the origin (Figure 3-1). If we rotate the isotropic pattern antennas around the z-axis, the problem remains unchanged, which means that all great-circle (constant φ) patterns are identical. On the z-axis, the element phase constant becomes ej kz cos θ . For simple line arrays we can locate pattern nulls and peaks by simple arguments. Example Two elements are spaced λ/2 and have equal amplitudes and phases. Locate the nulls and peaks. The phase reference planes can be placed at any convenient point. Consider the pattern at θ = 90◦ . We place the reference plane through the axis of the array. The added phase factor is zero for both elements and we just add components. The equal element phases add to give a beam peak. If we place a second reference plane through the top element, the wave radiated from the bottom element travels across the array λ/2 to the reference plane. Increasing the distance propagated decreases phase and it changes by −180◦ . The two out-of-phase signals cancel to produce a pattern null. The array has symmetry about the x –y plane, which means that the array will have the same pattern above and below the symmetry plane. We denote this conﬁguration an even-mode array. Figure 3-2 plots this pattern with a solid line. You should repeat the example for an odd-mode array (phases 0◦ and 180◦ ) and convince yourself that the null occurs at θ = 90◦ and the beam peak occurs at θ = 0◦ (180◦ ), plotted in

TWO-ELEMENT ARRAY

105

z d/2 d cos q 2 Radiation

q

Reference plane

−d cos q 2 −d/2

FIGURE 3-1

Two-element array on a z-axis.

Figure 3-2 as a short-dashed curve. The solid and short-dashed curves have the same directivity. Example Suppose that the two elements are spaced λ/4, with the top element phase −90◦ and the bottom element phase 0◦ . Locate the beam peak and pattern null. We start by placing a reference plane through the top element. The wave radiated from the bottom element travels across the array, and its phase decreases by 90◦ . Both radiated waves have the same phase (−90◦ ) at the reference plane and add in phase for a beam peak. Consider a second plane through the bottom element. The wave from the top element loses 90◦ propagating across the array and the two waves are 180◦ out of phase and cancel for a null. The second example is an end-ﬁre array. Figure 3-2 illustrates the end-ﬁre pattern with a long-dashed curve. All three patterns on the ﬁgure have the same directivity. The phase distribution of an end-ﬁre array matches those of a wave traveling in the direction of the maximum. In these examples unequal amplitudes would limit the null depth to the difference. Varying the element phases while maintaining equal amplitudes changes the null directions. Consider a general two-element array with equal amplitudes and a phase difference between them. We split the phase shift into equal parts. The top-element phase is −δ/2 and the bottom-element phase is δ/2. When we apply Eq. (3-2) with an isotropic element pattern, we obtain the following electric ﬁeld using Euler’s identity:

δ πd E(θ ) = 2E0 cos cos θ − λ 2

e−j kr r

(3-3)

106

ARRAYS

Even Mode l/2 Spacing

Odd Mode l/2 Spacing

Endfire l/4 Spacing

FIGURE 3-2 Two isotropic element array pattern: even-mode λ/2 spacing (solid curve); odd-mode λ/2 spacing (short-dashed curve); end-ﬁre λ/4 spacing (long-dashed curve).

θ is measured from the z-axis. If we spaced the elements along the x-axis and found the pattern in the x –z plane, we substitute sin θ for cos θ in Eq. (3-3). In Chapter 4 we sample continuous distributions and position the elements along the x- or y-axis. Pattern peaks occur when the argument of the cosine is nπ, the nulls when it is (2n − 1)π/2. δ λ cos θmax = nπ + (3-4) 2 πd π δ λ cos θnull = (2n − 1) + (3-5) 2 2 πd If we subtract either Eq. (3-4) or (3-5) evaluated at two peaks or nulls, we get the same equation: λ cos θ1 − cos θ2 = (n1 − n2 ) (3-6) d Figure 3-3 illustrates the pattern of an equally phased two-isotropic-pattern-element array spaced 5λ along the z-axis. Because array symmetry makes the patterns on the right and left sides the same, we consider only one side. The wide element spacing

TWO-ELEMENT ARRAY

107

5l Spaced Array

Array - 10 dB Added to Isotropic

FIGURE 3-3 Two-isotropic-element array-spaced 5λ pattern (solid curve); added central element 10 dB higher power than array (dashed curve).

allows six solutions to Eq. (3-4) from 0 to 90◦ for the pattern peaks and ﬁve solutions for Eq. (3-5) over the same range for the nulls because the magnitude of cos θ is limited to 1. We call the multiple beams grating lobes. We usually choose the main beam and call the others grating lobes, but they are just all lobes of the array. Figure 3-3 shows that we must space the elements close together to prevent grating lobes. With a greater number of elements in the array, the amount of beam movement due to element phasing adds another factor to the prediction of when grating lobes form. The amount of phase scanning determines the maximum spacing allowed without the formation of grating lobes. The n = 0 lobe forms at θ = 90◦ and we compute the n = 1 mode direction from Eq. (3-4): θ = cos−1 ( 15 ) = 78.46◦ . When we substitute these angles into Eq. (3-3), we calculate a relative phase of 180◦ between them. The lobes have a phase of zero for n even and 180◦ phase for n odd in the far-ﬁeld approximation. Remember we remove the exponential and 1/R factors from Eq. (3-3) for the far-ﬁeld pattern. The actual phase of any real point depends on the distance from the center of the array. The dashed curve in Figure 3-3 shows what happens if we add the array pattern to an isotropic radiator in the center. For a peak response of the array −10 dB relative to the isotropic antenna, we get the 5.7-dB peak-to-peak ripple shown by using Scale 1-8. The array pattern either adds or subtracts from the isotropic radiator pattern. The angular ripple rate is half that of the array lobes. Below we see that a two-element array

108

ARRAYS

spaced at an integer multiple of λ/2 has a 3-dB greater gain than a single element. We feed half the power of the array into each element. By adding these factors we calculate the array element level to be −16 dB below the main central radiating antenna. When we mount an antenna over a ﬁnite ground plane, the diffraction from the edges creates a two-element array. A 5λ-wide ground plane would produce the same pattern ripple angular rate as shown in Figure 3-3. You will often observe a similar-amplitude ripple in measured antenna patterns. Note the minimum angular distance between the peak and minimum responses in the pattern. The extra signals occur along the line in the pattern plane perpendicular to this direction. Use Eq. (3-6) to determine the distance between the array elements and you should be able to identify the structure causing the ripple. The scattering point could be on the test ﬁxture. Consider whether the mounting structure will be different in the ﬁnal conﬁguration. You can calculate the effect from a single diffraction point by forming an array using the baseline of the primary radiator and the diffraction point. Both conﬁgurations produce the same angular ripple rate. The ripple peak occurs along that array axis, but Figure 3-3 shows that the angular ripple rate will be reduced along this end-ﬁre direction of the θ = 0 axis. If you make a careful consideration of the angular rates, in various pattern planes, you should be able to discover the cause. Always consider unexpected sources of diffraction. You can consider the ripple using its beamwidth. To produce a symmetrical pattern about zero, we use sin θ instead of cos θ in Eq. (3-3), which means that the array lies along the x-axis. The −3-dB angle for the two-element uniform amplitude array can be found from Eq. (3-3): πd π sin θ3 dB = λ 4

θ3 dB = sin−1

λ 4d

(3-7)

The beamwidth is twice Eq. (3-7). For large d we can approximate sin X ≈ X and beamwidth = λ/2d. The 5λ spaced array has a beamwidth of 5.7◦ (0.1 rad). We can look at a 5λ-wavelength ground-plane example that has a large-amplitude element compared to the edge diffraction as two 2.5λ-spaced two-element arrays where one element has a high amplitude. Each two-element array produces a pattern with an 11.4◦ beamwidth the value of the composite pattern in Figure 3-3. We often mount an antenna in the center of a ground plane for measurement and observe patterns similar to Figure 3-3. If in the actual application the antenna is mounted off center, we need to add the patterns of arrays formed on both sides of the ﬁnite ground plane. The ﬁnal pattern will be the composite pattern from each array and be more complicated than the simple case given above. We calculate average radiation intensity by an integral: Uavg = The directivity is

4E02 η

π/2 0

cos2

δ πd cos θ − sin θ dθ λ 2

|2Emax |2 Umax = Uavg 1 + sin(2πd/λ) cos δ/(2πd/λ)

(3-8)

where Emax = cos[(πd/λ) cos θmax − δ/2]. When d ≥ λ/2, Emax = 1. Figure 3-4 shows the directivity versus spacing for the special cases δ = 0◦ and δ = 180◦ (even and odd modes). The directivity varies because each antenna receives power from the other. The

LINEAR ARRAY OF N ELEMENTS

109

Odd-mode array (0°, 180°)

4

Directivity, dB

3

2

Even-mode array (0°, 0°)

1

0.5

1.0

1.5 Spacing, l

2.0

2.5

FIGURE 3-4 Directivity of even- and odd-mode two-isotropic-element arrays.

combination of the input power and the power transferred between elements changes with spacing. 3-2 LINEAR ARRAY OF N ELEMENTS Suppose that there are N isotropic radiators equally spaced along the z-axis and fed with equal amplitudes. We assign a ﬁxed phase shift δ between progressive elements. The array factor ﬁeld is sin(N ψ/2) (3-9) N sin(ψ/2) where ψ = kd cos θ + δ [1, p. 258]. We use this to plot a universal radiation pattern for the array (Figure 3-5) for two to 10 elements. The abscissa ψ is plotted in degrees (360◦ is substituted for 2π in k). Both ends of the plot are lines of symmetry. The plot is periodic (period 360◦ ). We see that the level of the ﬁrst sidelobe (N = 2 has no sidelobe) decreases as N decreases but approaches a limit of 13.3 dB of the continuous aperture. Figure 3-6 demonstrates the periodic pattern for N = 6 and shows a projection to a polar pattern when the progressive phase between elements is zero and the elements are spaced λ/2. We can plot similar curves for other array distributions; all have a period of 360◦ . Figure 3-6 illustrates the use of a circle diagram, a method of constructing a polar pattern from the universal pattern such as Eq. (3-9) for the uniform-amplitude distribution. An array can be analyzed as a sampling of the continuous distribution that produces a Fourier series of the distribution. A Fourier series has multiple responses. In Chapter 4 we design large arrays by sampling continuous distributions. The pattern angle of an array is measured either from the axis using cosine of pattern angle or is measured broadside using sine. You should become comfortable with either notation since the sine and cosine of angles involves only a complementary operation of the angles.

110

ARRAYS

−2 −4

3

4

N=2

−6 Amplitude, dB

−8 6

−10

3 4

8

−12

8 10

−14

6

5

10

5 7

−16 8

−18 −20

FIGURE 3-5

6

0

20

40

60

80 100 120 kd cos q + d

140

160

180

ψ-space pattern of linear arrays with a uniform amplitude distribution.

360 sin q Spacing/l Visible Region

FIGURE 3-6 Circle diagram of a six-element uniform-amplitude array with λ/2 spacing.

Since cos θ (or sin θ ) is limited to ±1, the region along the abscissa of the universal pattern used (the visible region) is found from the range of ψ: −360◦ d +δ λ

to

360◦ d +δ λ

LINEAR ARRAY OF N ELEMENTS

111

The circle diagram is constructed by ﬁrst drawing a circle the same diameter as the visible region below the universal diagram centered at δ, the progressive phase shift between elements. Figure 3-6 has a δ = 0. Since the element spacing is λ/2, the range is ±180◦ . The polar pattern radius equals the amplitude of the universal pattern. Both the universal pattern and the polar pattern use a logarithmic (dB) scale from 0 to −40 dB. Projecting points vertically from the universal pattern to the visible region performs the cosine or sine operation, and the polar pattern becomes the real pattern in space. We project each point vertically until it intersects the dashed visible region circle in two places and then draw lines from these points to the center. After you project the nulls and peaks of the universal pattern to the dashed circle, it is easy to sketch the polar pattern. The circle diagram helps us visualize patterns and the effects of scanning, but no one would do serious design with it. Second, it is useful only for small arrays because large arrays produce unwieldy diagrams. When the spacing between elements is greater than λ/2, the visible region widens to include more than one periodic main lobe and the array has multiple beams. To have a beam centered at θ1 , set the progressive phase difference between elements: δ=

−360◦ d cos θ1 λ

(3-10)

−360◦ d λ

(3-11)

End ﬁre (θ1 = 0) occurs when δ=

We can use Figure 3-5 to compute beamwidth angles of arrays. Table 3-1 is a list of the ψ-space angles of the 3- and 10-dB levels. Example A six-element equally spaced uniform array has spacings of λ/2 and zero progressive phase shift between elements (δ = 0◦ ). Calculate the 3-dB beamwidth. We read from Table 3-1 the value ψ3 dB = 26.90◦ . Because the pattern is symmetrical in ψ space (Figure 3-6), the second ψ3 dB is −26.9◦ . kd cos θ1,2 + δ = ±ψ3 dB ±26.9◦ 360◦ λ ◦ cos θ1,2 = cos θ1,2 = ±26.90 λ 2 180◦ ◦ ◦ θ1 = 81.4 θ2 = 98.6 TABLE 3-1 ψ-Space Angles of 3- and 10-dB Levels of an Equal-Amplitude Distribution Array (deg) N

3 dB

10 dB

N

3 dB

10 dB

N

3 dB

10 dB

2 3 4 5 6 7 8 9 10

90.00 55.90 40.98 32.46 26.90 22.98 20.07 17.81 16.02

143.13 91.47 67.63 53.75 44.63 38.18 33.36 29.62 26.64

11 12 13 14 15 16 17 18 19

14.55 13.33 12.30 11.42 10.65 9.98 9.39 8.87 8.40

24.21 22.18 20.47 19.00 17.74 16.62 15.64 14.77 14.00

20 24 28 32 36 40 50 64 100

7.980 6.649 5.698 4.985 4.431 3.988 3.190 2.492 1.595

13.29 11.08 9.492 8.305 7.382 6.643 5.314 4.152 2.657

112

ARRAYS

Remember that θ is measured from the axis of the array (z-axis) and the 3-dB beamwidth is the difference (17.2◦ ). On Figure 3-6 the visible region ranges between −180◦ and +180◦ . There are four sidelobes in the visible region (Figure 3-6). Since an array samples a continuous aperture distribution, the continuous distribution is Nd long. We can estimate beamwidth by using a uniform amplitude distribution: ◦

HPBW = 50.76

λ ◦ = 16.92 Nd

This formula approximates the array beamwidth reasonably. Example A six-element array has a progressive phase shift δ of 90◦ between elements. Compute the 10-dB beam edge angles for λ/2 spacing. Figure 3-7 shows the circle diagram analysis of this example. The line to the center of the polar pattern has been shifted to 90◦ and the pattern spans 360◦ of the linear scale. By projecting the nulls and peaks to the circle below, the pattern can easily be sketched. ◦

ψ10 dB = ±44.63

(Table 3-1)

kd cos θ1,2 = ±ψ10 dB − δ

360 sin q Spacing/l Phase Shift

Visible Region

FIGURE 3-7 Six-element uniform-amplitude array with λ/2 spacing scanned with 90◦ progressive phase shift between elements.

LINEAR ARRAY OF N ELEMENTS

113

Solving for cos θ1,2 , we have ±ψ10 dB − δ ±44.63◦ − 90◦ = kd 180◦ ◦ ◦ ◦ θ1 = 104.6 θ2 = 138.4 beamwidth = 33.8

cos θ1,2 =

There are ﬁve sidelobes in the visible region (Figure 3-7). Equation (3-9) gives the beam maximum direction: cos θ0 =

−δ −90◦ = kd 180◦

◦

θ0 = 120

The main beam is no longer symmetrical about the beam peak. The 3-dB pattern angles are 110.5◦ and 130.5◦ . The beamwidth (3-dB beamwidth = 20◦ ) increases with scan angle. What element spacing would result in this beamwidth for broadside radiation (δ = 0◦ )? 360◦ ◦ d cos θ1 = 26.90 (Table 3-1) λ On solving for spacing, we have d 26.9◦ = λ 360◦ cos θ1 Remember that the beam is centered on θ = 90◦ , so that θ1 = 90 − 20/2 = 80◦ . 26.9◦ d = = 0.431 λ 360◦ cos 80◦ The effective spacing has been reduced by approximately the cosine of the scan angle from θ = 90◦ , broadside: d ◦ cos 30 = 0.433 λ The accuracy of the cosine relation increases with more elements. Example Determine the progressive phase shift between elements for an end-ﬁre array with 0.3λ element spacing and compute beamwidth for a uniform distribution array with ﬁve elements. Figure 3-8 illustrates this example using the circle diagram. End ﬁre occurs when [Eq. (3-11)] −360◦ (0.3λ) ◦ δ= = −108 λ This is the progressive phase shift for all distributions with 0.3λ element spacing for an end-ﬁre pattern. Table 3-1 gives the ψ-space angle, ψ3 dB = ±32.46◦ . Substituting in the expression for ψ, we have 360◦ (0.3λ) ◦ cos θ1,2 = ±32.46 λ ±32.46◦ + 108◦ cos θ1,2 = 360◦ (0.3) 140.46 cos θ1 = = 1.301 108

cos θ2 =

75.46 = 0.699 108

114

ARRAYS

Visible Region

360 sin q Spacing/l Phase Shift

FIGURE 3-8 Five-element uniform-amplitude array scanned to end ﬁre.

θ1 is in invisible space, since | cos θ | ≤ 1; θ2 = 45.6◦ . Symmetry about the z-axis supplies us with the second angle θ1 = −45.6◦ and beamwidth is the difference: 91.2◦ . The end-ﬁre array samples a traveling-wave distribution. The continuous uniform distribution phased for end ﬁre with the same length has a 90.4◦ beamwidth. Remember that we have been dealing with isotropic pattern antennas. For example, broadcast towers, seen from above, approximate isotropic antennas in the horizontal plane. The patterns of the individual antennas modify the results of isotropic antenna arrays. In small arrays the element pattern is quite signiﬁcant, but the beamwidths of large arrays are determined mainly by the array factor. The beamwidths calculated for array factors approximate the actual beamwidths only when the elements have signiﬁcant patterns. We must rely on computer solutions of speciﬁc cases, including the element pattern, for better results. 3-3 HANSEN AND WOODYARD END-FIRE ARRAY [2] The end-ﬁre array directivity increases if the sum of the progressive phase shifts between elements is decreased by approximately π. The equivalent traveling-wave velocity slows in the structure relative to free space. The progressive phase shift between elements becomes δ = −kd −

2.94 π −kd − N N

rad

(3-12)

PHASED ARRAYS q

115

q Hansen and Woodyard end fire

Normal end fire

10 dB

20 dB

30 dB

FIGURE 3-9 Patterns of a normal end ﬁre and a Hansen and Woodyard end-ﬁre array of isotropic elements.

where N is the number of elements in the array. The normal end-ﬁre progressive phase shift between elements, δ = −kd, places one edge of the visible region at the origin of ψ space. This method shifts the edge to a lower portion of the curve. The universal radiation curve peak (Figure 3-5) shifts into invisible space and the sidelobes rise in proportion to the new beam peak, but the beamwidth narrows. Equation (3-12) holds strictly only for large arrays, but the directivity increases for all arrays when it is applied. Example Suppose that eight elements are spaced λ/4 apart with a uniform amplitude distribution. Compare the two endﬁre designs. The two patterns are compared in Figure 3-9. The results are as follows: The beamwidth decreases, and the directivity increases by 2.5 dB. The sidelobes rise to 9 dB from 13 dB.

3-4 PHASED ARRAYS Suppose that a wave approaches at an angle to the axis of an array located on the z axis (Figure 3-10). The wave reaches the top element ﬁrst and progresses down the array in succession. If the signals are added directly, they will cancel each other to some extent because they have a progression of phases. Figure 3-10 shows the results of adding a series of time delays to equalize the path lengths in the lines where the position zi along the axis determines the time delay τi for incident angle θ0 : τi =

zi cos θ0 + τ0 c

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ARRAYS

t1 Time delay t2

t3

t4

t5

Incident wave

t6

FIGURE 3-10 Linear array scanned with time-delay networks.

and velocity of light c. We add an arbitrary time delay τ0 to keep all time delays, τi , positive. This feed network is frequency independent, as we vary the progression of time delays to scan the beam. Phase shifters replace the time-delay networks in phased arrays. They provide equivalent beam scanning at a single frequency. To scan to an angle θ0 , the required phase shift is 2π − z cos θ0 modulo 2π (rad) λ −360◦ ◦ z cos θ0 modulo 360 (deg) λ for elements located on the z-axis. For a general space array we must counteract the phase difference to the reference plane, ej k·r , for the direction of scan so that the phases of all elements are zero. To scan in the direction (θ0 , φ0 ), we must add a phase factor to every element, depending on its position. The phase factor on each element of a general space array is e−j k0 · r (3-13) where k0 =

2π (sin θ0 cos φ0 xˆ + sin θ0 sin φ0 yˆ + cos θ0 zˆ ) λ

GRATING LOBES

117

is the vector propagation constant in the direction of the beam and r is the element location. Adding this phase factor to the element phases causes the product of the exponential factors [Eq. (3-2)] to be 1 at the scan angle, and the components Ei add in the scan direction. Using phase shifters limits the frequency bandwidth. Given a ﬁxed phase shift over a small frequency range, increasing the frequency scans the beam toward broadside: θ =

π f2 − f1 − θ0 tan f2 2

(3-14)

rad

where θ0 is the scan angle [3]. Limiting the allowable scanning with frequency to plus or minus one-fourth of the local beamwidth deﬁnes the bandwidth of the array. When the beam is scanned to 30◦ off the axis, the bandwidth is related directly to the beamwidth at broadside (θ = 90◦ ): ◦

bandwidth(%) beamwidth (deg) at θ0 = 30

The beam shifts less with frequency near broadside, since the tangent factor in Eq. (314) approaches zero. A general estimate is given by bandwidth(%)

beamwidth (deg) 2 cos θ0

(3-15)

where the broadside beamwidth is used. Example Given an array with 100 elements spaced at λ/2, determine the bandwidth when scanned to 45◦ . The beamwidth is estimated from the aperture width: HPBW =

50.76◦ ◦ 1 100( 12 )

bandwidth(%)

1 = 0.7% 2 cos 45◦

Any radar antenna would have a broader beamwidth because the sidelobes need to be reduced, but this is a good ﬁrst estimate. The bandwidth can be increased by feeding subarrays with time-delay networks. The subarrays continue to be scanned with phase shifters. Only a few time-delay networks are needed, and the subarray beamwidth determines the bandwidth. In Chapter 12 we discuss the problems caused by using subarrays. 3-5 GRATING LOBES Phased arrays vary the progressive phase by Eq. (3-13) to scan the beam. When the array element spacing is greater than λ/2, the appearance of secondary beam peaks (grating lobes) limits the scan angle. The grating lobe attains full amplitude when d −1 λ (1 + cos θgr ) = 1 −1 (3-16) θgr = cos λ d

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ARRAYS

360 sin q Spacing/l Phase Shift

Visible Region

FIGURE 3-11 Ten-element array with 3λ/4 spacing scanned to 26◦ , showing the onset of a grating lobe.

Example The spacing of the elements of an array is 0.75λ. Determine the scan angle when the grating lobe is full amplitude. 4 ◦ − 1 = 70.5 θgr = cos−1 3 At this point the grating lobe is the same amplitude as the main beam. The lobe does not appear suddenly, but it grows as the visible region shifts and starts including the second periodic main lobe. Figure 3-11 shows the grating lobe formation for an array with 0.75λ element spacings on a circle diagram. The dashed circle of the visible region spans more than one beam of the universal radiation pattern of the uniform amplitude array. Arrays with element spacing greater than λ always have grating lobes (multiple main beams), but the pattern of the antenna elements may reduce the grating lobes to acceptable levels and allow a wide element spacing. 3-6 MULTIPLE BEAMS An array can form multiple beams. Equation (3-13) gives the phase coefﬁcients to multiply each element feed voltage Ei to scan it to a given angle. The array will form

MULTIPLE BEAMS

119

a second beam if we add a second distribution: Ei e−j k2 ·ri . The distribution Ei remains constant for both beams. We add the two distributions to obtain both beams:

Ei (e−j k1 ·ri + e−j k2 ·ri )

(3-17)

This multiplies the distribution Ei by a second distribution whose amplitudes and phases are functions of the antenna position and the scan angles of the two beams. Each beam uses the entire array to form its beam. In a phased array both phase and amplitude must be varied to achieve multiple beams. An array, which can only vary phase, must be divided into subarrays to form multiple beams, but its beamwidths will depend on the subarray widths. We can produce unequal beams with different amplitude distributions and pattern shapes if needed. We can add as many beams as necessary by including the distribution element factors with the scanning phase coefﬁcients in Eq. (3-17). The element feeding coefﬁcients become the sum. Example Compute the feed coefﬁcients of a 15-element array with λ/2 spacings and a uniform distribution scanned to 45◦ and 120◦ from the z-axis. First center the array on the z-axis. The elements are located at zi =

(−8 + i)λ 2

To scan to 45◦ , the element phase factors are 360◦ 1 (−8 + i)λ exp(−j kzi cos 45 ) = exp −j √ λ 2 2 ◦

◦

phase factors are ej 90 (−8+i) . The ninth-element (z9 = λ/2) To scan to 120◦ , the element ◦ ◦ −j 127.3 j 90 phase factors are e and e . Assume a voltage magnitude of one-half for each uniform-amplitude-distribution beam so that the center element has a magnitude of 1. We sum◦ the distributions to calculate the feeding coefﬁcient of the ninth element:0.32ej 161.4 . When converted to decibel ratios, Table 3-2 lists the feeding coefﬁcients for the array. We can estimate both beamwidths from Table 3-1. ψ3 dB = 10.65◦ :

cos θ1,2

 ±10.65◦ + 127.28◦    180◦ =  ±10.65◦ − 90◦   180◦

θ1 = 40◦ , θ2 = 49.6◦ θ1 = 116.2◦ , θ2 = 124◦

The pattern (Figure 3-12) has these beams. The gain of each beam depends on the feed network. If a single input supplies power to two beams, each beam can receive only half the input power and gain reduced 3 dB for both beams. Butler matrices [4] and Blass beamforming networks [5] supply an input for each beam. The inputs are isolated from each other and the transmitter power in each port feeds only one beam, and therefore the full array gain is available to each input. Similarly, we can place a receiver on each port and use the full effective area for each.

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ARRAYS

TABLE 3-2 Feeding Coefﬁcients for a Dual-Beam 15-Element Array Beamwidth (dB)

Angle (deg)

Element

Element

Beamwidth (dB)

Angle (deg)

1 2 3 4 5 6 7 8

−2.38 −8.59 −0.01 −11.49 −1.64 −1.99 −9.91 0.00

130.48 111.84 −86.80 74.56 55.92 −142.72 161.36 0.0

9 10 11 12 13 14 15

−9.91 −1.99 −1.64 −11.49 −0.01 −8.59 −2.38

161.36 142.72 −55.92 −74.56 86.80 −111.84 −130.48

FIGURE 3-12 Fifteen-element linear array pattern with simultaneous beams at θ = 45◦ and 120◦ .

We will delay the important topics of array synthesis and sidelobe reduction until after we have discussed aperture distributions. A trade-off is made between the beamwidth and the sidelobe levels. The beamwidth narrows only by putting more power into the sidelobes. 3-7 PLANAR ARRAY The linear array only controls the pattern in one plane; it depends on the element pattern to control the beam in the other plane. Planar arrays can control the beam shape in both planes and form pencil beams. Whereas a linear array can only scan in

PLANAR ARRAY

121

a single plane, a planar array can scan to any angle in the upper hemisphere. Most planar arrays rely on the element pattern or ground plane to eliminate the backlobe on the opposite side of the plane. The planar array has N − 1 nulls that can be used to control the pattern where N is the total number of elements. A simple feed distribution uses the product of two linear arrays. This eliminates many degrees of freedom of the array because an M × N array would be determined by M − 1 + N − 1 nulls when we could have used M × N − 1 possible nulls. Figure 3-13 shows the spherical pattern of a uniformly spaced 8 × 8 planar array where all elements are fed the same amplitude where a 90◦ beamwidth element eliminates the backlobe. The pattern along either principal axis shows the steady sidelobe reduction that starts with −13.2 dB. Diagonal plane sidelobes are the product of the sidelobes in the principal planes. The ﬁrst sidelobe in the diagonal plane is down 26.4 dB. An array feed distribution, not a product of two linear arrays, can yield more equal sidelobes in all pattern planes. Figure 3-14 illustrates the pattern of the rectangular array when the element feeding coefﬁcients are phased to scan the beam along one principal plane. The main beam broadens in the plane of scan as the effective array length is reduced but stays narrow in the orthogonal plane. More sidelobes appear behind the main beam. We see a large sidelobe growing on the horizon that will become a grating lobe when the array is scanned further. The sidelobes in the plane orthogonal to the plane of scan move with the main beam but roll into a cone that becomes tighter with increased scan.

FIGURE 3-13 Spherical radiation pattern of an 8 × 8-element uniform-amplitude and spaced square planar array.

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ARRAYS

FIGURE 3-14 Spherical radiation pattern of an 8 × 8-element square-planar array scanned along a principal plane. 2.5 2 1.5 1

ky-space

0.5 0 −0.5 −1 −1.